Norman attended Greenwood High School, where he was a teammate of other future NFL players in Armanti Edwards and D. J. Swearinger. In 2006, Norman was the only two-way player on his high school team that won the South Carolina State Championship. He was named an All-State football player and was selected to participate in the North-South All-Star Game. He was also named to the All-Lakeland Region 4A on offense and also participated in basketball, baseball, and track throughout high school. Despite all that, Norman received no scholarship offers from any Division I schools.

Norman began college by taking courses at Horry Georgetown Tech while living with his brother, Marrio Norman, who earned a full scholarship in football to Coastal Carolina. The following year in 2008, Norman was offered a spot on the Chanticleers as a walk-on. He eventually graduated with a communications major and dramatic arts minor.

During his true freshman year at Coastal Carolina in 2008, Norman played in all 12 games with 7 starts. During his first start against North Carolina A&T, Norman recorded his first interception. He recorded 26 solo tackles and 35 total tackles, tying for ninth most tackles on the team. He ranked fifth in the Big South with nine pass defenses and two interceptions. He earned Big South Freshman of the Week honors after his seven tackle and one interception performance in a win over Presbyterian. After his freshman year, Coastal Carolina awarded Norman a scholarship.

During his junior season, Norman again earned 1st team All-Big South. He ranked fifth on the team with 56 tackles and 3rd in the Big South with 9 pass defenses. He also recorded another blocked field goal, bringing him one shy of the Coastal Carolina record. In 2011, Norman was a preseason All-American and All-Big South conference as well as being listed as one of the top 10 defensive backs for 2011 by The Sporting Network. Additionally, he was voted onto the 2011 AFCA Football Championship Subdivision Coaches’ All-America Team. For his college career, Norman tied for second in Coastal Carolina history with 11 career interceptions and pass breakups with 23.

On January 12, 2012, it was announced that Norman had accepted his invitation to play in the East-West Shrine Game. Norman was projected to be a late round pick at the end of his senior season, but began to climb draft boards after impressing scouts during East-West Shrine Game practices. On January 21, 2012, Norman played in the 2012 East-West Shrine Game and was part of the East team that lost 24-17 to the West. On January 26, 2012, Norman accepted an invitation to play in the 2012 Senior Bowl as a late-replacement after Alfonzo Dennard sustained an injury. Norman continued to impress coaches and scouts by intercepting six passes during Senior Bowl practices, although he was a late-replacement and had minimal time to practice with his team. On January 28, 2012, he played in the 2012 Senior Bowl as part of Minnesota Vikings’ head coach Leslie Frazier’s North team that defeated the South 23-13.

He attended the NFL Scouting Combine in Indianapolis and completed all of the combine drills. His mediocre performance in the 40-yard dash (4.66s) possibly resulted in a fall in his draft stock. On March 27, 2012, Norman participated at Coastal Carolina’s pro day and chose to perform combine and positional drills. He improved his time in the 40-yard dash (4.57s), 20-yard dash (2.63s), and three-cone drill (6.92s). Norman attended a pre-draft visit with the Detroit Lions. It was reported that team representatives and scouts from 27 NFL teams attended Coastal Carolina’s pro day. At the conclusion of the pre-draft process, Norman’s draft projections varied from as early as the third round to as late as the seventh round by NFL draft experts and scouts. The majority of NFL draft experts projected Norman to be a third or fourth round pick. He was ranked the eighth best cornerback prospect in the draft by NFL analyst Mike Mayock, 10th best by Scouts Inc., 28th best CBS Sports, and 29th best by DraftScout.com.

The Carolina Panthers selected Norman in the fifth round, 143rd overall, of the 2012 NFL Draft. Norman was the 14th cornerback drafted in 2012 and became only the third player to be drafted from Coastal Carolina, after Tyler Thigpen (2007) and Jerome Simpson (2008).

On May 10, 2012, Norman signed his four-year rookie contract with the Panthers, worth US$2.3 million. Throughout training camp, Norman competed to be a starting cornerback against Captain Munnerlyn and Brandon Hogan. Head coach Ron Rivera named Norman and Chris Gamble the starting cornerbacks to begin the regular season.

He made his professional regular season debut and first career start in the Panthers’ at the Tampa Bay Buccaneers and recorded four combined tackles during their 16–10 loss. In Week 3, he collected a season-high 11 combined tackles (eight solo) and a pass deflection during a 36–7 loss to the New York Giants. Norman became a focal point of the secondary after the Carolina Panthers placed Chris Gamble on injured reserve due to a shoulder injury on October 22, 2012. On October 28, 2012, Norman recorded five combined tackles, deflected a pass, and made his first career interception in the Panthers’ 23-22 loss at the Chicago Bears in Week 8. Norman intercepted a pass by Bears’ quarterback Jay Cutler, that was originally intended for wide receiver Brandon Marshall, during the first quarter. Norman began to struggle towards the end of his rookie season and was eventually surpassed on the depth chart by Josh Thomas and was relegated to a backup cornerback beginning in Week 14. In Week 15, Norman was surpassed on the depth chart by James Dockery and became the fourth cornerback on the depth chart. He finished his rookie season in 2012 with 73 combined tackles (52 solo), seven pass deflections, and one interception in 16 games and 12 starts.

On September 15, 2013, Norman collected a season-high four combined tackles during the Panthers’ 24–23 loss at the Buffalo Bills. During the fourth quarter, Buffalo Bills’ quarterback E. J. Manuel threw a two-yard touchdown pass to a wide open Stevie Johnson after a miscommunication between Josh Norman and cornerback D. J. Moore. Moore stated he called for Norman to switch and cover Johnson as Moore covered wide receiver Chris Hogan. Norman stated he didn’t hear Moore call for a switch. As a result of the mistake, Norman was effectively benched for the majority of the season. Head coach Ron Rivera benched Norman for three games (Weeks 5–7) as a healthy scratch. He was demoted to being the fifth cornerback on the depth chart behind Captain Munnerlyn, Josh Thomas, Melvin White, and Drayton Florence. Norman was inactive for six of the next nine games (Weeks 9–10, 14–17) as a healthy scratch. Norman finished the 2013 NFL season with four combined tackles (three solo) in seven games and zero starts.

In Week 5, Norman earned his first start of the season, over Melvin White, and recorded three combined tackles in the Panthers’ 31–24 win against the Chicago Bears. Norman suffered a concussion during the game and was sidelined for the next two games (Weeks 6–7). In Week 8, Norman became a starting cornerback after surpassing Melvin White and Antoine Cason on the depth chart. On November 16, 2014, Norman collected a season-high seven combined tackles and deflected a pass during a 19–17 loss to the Atlanta Falcons in Week 11. His success facilitated the Panthers to release Cason on December 2, 2015. Norman began to show his potential as a shutdown corner in the second half of the 2014 season while continually covering the opposing team’s top receiver. In Week 10, Norman held Philadelphia Eagles’ wide receiver Jeremy Maclin to three catches for 38-yards and no touchdowns. The following week, he held Atlanta Falcons’ wide receiver Julio Jones to six receptions for 59-yards and no touchdowns. In Week 15, Norman held rookie Tampa Bay Buccaneers’ wide receiver Mike Evans to two catches for 13-yards and no touchdowns. On December 21, 2014, Norman recorded five combined tackles, deflected a pass, and made an interception during a 17–13 win against the Cleveland Browns in Week 16. He held Browns’ wide receiver Josh Gordon to four catches for 45-yards. Over the final eight weeks of the season, Norman was rated as the 27th best cornerback by Pro Football Focus and gave up only 13 receptions on 33 targets. He finished the season with 48 combined tackles (36 solo), 11 pass deflections, two interceptions, and a forced fumble in 14 games and ten starts. For the season, Norman was targeted 58 times, but only allowed 27 catches for 291 yards and just 74 yards after the catch. The 46.6% catch rate was second best in the NFL among starters behind only Vontae Davis.

The Carolina Panthers finished first in the NFC South with a 7–8–1 record and earned a playoff berth. On January 3, 2015, Norman started his first career playoff game and recorded four combined tackles as the Panthers defeated the Arizona Cardinals 27-16 in the NFC Wildcard Game. Norman gave up only one reception for nine-yards during the game. The following week, the Panthers were eliminated from the playoffs after a 31-17 loss at the Seattle Seahawks in the NFC Divisional Round.

In 2015, Norman picked up where he left off from the 2014 season. During the season opening victory against the Jacksonville Jaguars, Norman was thrown at nine times, allowed just three completions for five yards, and managed to force a fumble and collect a pick-six against quarterback Blake Bortles. Norman followed this up during the Panthers week 2 victory over the Houston Texans where he allowed just 40 yards in coverage on 12 attempts, no first down receptions, while helping to limit DeAndre Hopkins to five receptions and 53 yards for the game. The following week, Norman sealed a 27–22 win against the New Orleans Saints after an interception in the end zone on a pass by Luke McCown intended for Brandin Cooks with a minute remaining in the game. Through the first three weeks of the season, the only two cornerbacks who were targeted 20 or more times and had not allowed a pass for more than 16 yards were Norman and teammate Bené Benwikere. Further, through that span, Norman did not allow 50 yards receiving in a game and quarterbacks generated a 38.8 passer rating when throwing in his direction. For his efforts, Norman was named the NFC’s defensive player of the month for September. The following week, in a 37–24 victory over the Tampa Bay Buccaneers, Norman became the first cornerback since Charles Woodson in 2008 with 2 INT-TDs in the first 4 weeks of a season. Against the Buccaneers, Norman was targeted 8 times and allowed only two receptions for 31 yards, while recording 2 interceptions and 2 pass defenses. Further, Jameis Winston had a 3.6 quarterback passer rating when targeting Norman. He was named the NFC Defensive Player of the Week for his performance.

During the Panthers’ Week 15 game, which resulted in a 38–35 victory over the New York Giants, Norman had a high-profile contest with receiver Odell Beckham Jr.. During the game, Norman was targeted 6 times when matched up with Beckham and gave up 4 receptions for 30 yards and a touchdown. Tempers flared between the two players with Norman committing 2 personal fouls and Beckham committing 3 personal fouls, which included a hit to the head of a defenseless Norman that later resulted in a 1-game suspension for Beckham. Norman was selected for his first career Pro Bowl during the season. On December 23, 2015, Norman was fined a combined $26,044 for a face mask penalty, and for striking an opponent in the head against the Giants. During the final game of the regular season, Norman held Mike Evans to 15 yards from 1 catch on 5 targets.

On April 20, 2016, the Panthers rescinded their franchise tag that they placed on Norman, after he and the team failed to come to an agreement on a new contract, making him an unrestricted free agent.

On April 22, 2016, Norman signed a five-year, $75 million deal with the Washington Redskins. The contract made him the highest paid cornerback in NFL history at the time, surpassing the contract Darrelle Revis signed with the New York Jets in 2015. During the team’s training camp, Norman announced that he would appear regularly during that season’s Fox NFL broadcast. In Week 3, Norman faced off against Odell Beckham Jr. and the New York Giants, which generated media attention due to their previous controversial matchup in 2015. He recorded his first interception as a Redskin against quarterback Cody Kessler in the Week 4 win over the Cleveland Browns. In a Week 16 win against the Chicago Bears, he recorded two interceptions on quarterback Matt Barkley. He started in every game during the season, finishing with 67 tackles, 19 passes defended, two forced fumbles, and three interceptions. After the season, he was ranked as the 59th best player by his peers in the NFL Top 100 Players of 2017.

During his time with the Redskins, Norman appeared in several local car dealership commercials alongside Kirk Cousins and Ryan Kerrigan. In September 2017, Norman donated $100,000 to the recovery efforts following the devastation Puerto Rico received after Hurricane Maria impacted the country earlier that month.

In 2018, Norman appeared on season 26 of the American television show Dancing with the Stars. He was partnered with professional dancer Sharna Burgess, where he eventually progressed to the finals, finishing as the runner-up behind figure skater Adam Rippon. Norman also participated in a televised celebrity softball game during the 2018 Major League Baseball All-Star Game event week.

In Week 3 against the Chicago Bears, Norman sacked Mitchell Trubisky once and recorded his first interception of the season off a pass from Trubisky intended for Allen Robinson in the endzone as the Redskins lost 31-15. Due to poor performance, interim head coach Bill Callahan made the decision to replace Norman as a starter with Fabian Moreau for the Week 12 game against the Detroit Lions. Norman would play a reserve role taking snaps on special teams. He was released on February 14, 2020.

In March 2020, the Buffalo Bills signed Norman to a one-year, $6 million contract that includes $3 million guaranteed and a signing bonus of $1.50 million. He was placed on injured reserve on September 12, 2020, and was activated on October 3, 2020. In his debut with the Bills against the Las Vegas Raiders, Norman entered in placed of injured starter Levi Wallace. Norman forced a fumble on Raiders tight end Darren Waller in the fourth quarter and recovered the football himself, setting up a touchdown drive to seal the game as Buffalo won 30–23.

During the George Floyd protests in 2020, Norman and New Orleans Saints linebacker Demario Davis attended Buffalo mayor Byron Brown’s press conference announcing changes to the Buffalo Police Department following an incident between protesters and police in downtown Buffalo.