Born in Burnley, Lancashire, Norwood began his football career playing for his brother's team in his home town. He was spotted by Manchester United scouts when he was 6 years old and joined the club at the age of 7. After eight appearances for Manchester United's under-18s side in 2005–06 and 2006–07, he joined the club as a trainee in July 2007. He became a regular in the under-18s team in 2007–08, scoring twice in 22 appearances, and he made his debut for the reserve team in a 2–1 away win over Middlesbrough on 6 November 2007. The following season, he scored nine goals in 28 appearances for the under-18s, as well as appearing a further 10 times for the reserves. His form prompted the club to sign him on a professional contract in July 2009.
2009–10 was a big season for Norwood, as he made the full step up to the reserve team and also received his first call-up to the first-team for their Champions League group stage match away to Wolfsburg. However, although he made the trip to Germany, Norwood was not named in the 18-man match-day squad.
On 16 September 2010, Norwood signed for Carlisle United on a one-month emergency loan contract, following a knee injury to Carlisle captain Paul Thirlwell. He made his debut for the Cumbrian side on 18 September 2010 in a 0–0 draw at home to Brighton & Hove Albion, playing the full 90 minutes of the game. Norwood made a total of seven appearances during his loan spell, with six coming in the league, before his stay was cut short by injury in October 2010.
On 24 August 2011, Norwood signed for Scunthorpe United on a six-month loan. He made his debut for Scunthorpe in the side's 2–1 League Cup second round defeat at home to Newcastle United on 25 August 2011, when he played the full 90 minutes plus extra time. His league debut came just two days later with another start in a 3–2 defeat away at Sheffield Wednesday on 27 August 2011. His first league goal of the season came in Scunthorpe's draw at home to Huddersfield Town on 25 October 2011, scoring the opener on 10 minutes; the game ended 2–2. Norwood received the first red card of his career in Scunthorpe's 3–2 away defeat to Notts County on 26 November 2011, when he was sent off for a foul on Jeff Hughes. Norwood played a total of 16 starts and scored once during his loan spell before returning to parent club Manchester United in January 2012.
Norwood completed a loan switch to Championship side Coventry City on the January 2012 deadline day, and remained on loan until the end of the season. Norwood made his Coventry City debut in the team's 3–2 home defeat to Ipswich Town on 4 February 2012, coming on as a 25th-minute substitute for David Bell; his first start for the club came in their next game: a 2–0 defeat on 11 February 2012 away to Reading. On 21 March 2012, he scored a stoppage-time equaliser against Cardiff to earn a point for the Sky Blues. He scored his second Coventry City goal in the 2–0 home win against Portsmouth on 24 March 2012.
At the end of the 2011–12 season, Norwood confirmed that he was rejecting Manchester United's offer of a contract extension to pursue first-team football. In June 2012, Manchester United agreed compensation packages for Norwood with Championship clubs Barnsley and Huddersfield Town. Norwood rejected Barnsley's offer on 29 June, and he signed a three-year contract with Huddersfield later that day. He made his league debut in a 1–0 away defeat to Cardiff City on 17 August. Norwood scored his first goal for the Terriers with the opener in a 3–1 win over Sheffield Wednesday at Hillsborough Stadium on 19 September.
On 19 August 2014, Norwood was spotted in the crowd at the Madejski Stadium watching his current club, Huddersfield Town, playing the club he would soon join. He had his medical the following day, and then agreed personal terms the day after that. The fee was described by Huddersfield as a "significant and undisclosed fee". He took the number 6 shirt. Norwood made his Reading debut as a second-half substitute in the 4–0 defeat by Nottingham Forest on 23 August 2014.
On 3 August 2016, Norwood signed a three-year contract with Brighton & Hove Albion. On 16 August 2016, Norwood made his debut in the 3–0 win over Rotherham United.
During July 2017, Norwood joined EFL Championship side Fulham on a season-long loan deal following Brighton gaining promotion to the Premier League. He scored his first goal for Fulham in a 2-1 loss at Burton Albion on 16 September 2017.
Although he was born in England, Norwood's heritage is Northern Irish; it is this heritage that has allowed him to represent both England and Northern Ireland in international football. He was first called up to the England under-17 team in September 2007 ahead of their 2008 UEFA Under-17 Championship qualifying matches against Malta, Estonia and Portugal in Estonia from 21 to 26 October 2007. He appeared in all three fixtures, his debut coming in the 6–0 win over Malta on 21 October, in which he came on as a 61st-minute substitute for Harry Forrester. However, the match against Portugal would prove to be his final appearance for an England team.
Norwood made his first appearance for a Northern Ireland team at "B" international level, in a 3–0 defeat away to Scotland B on 6 May 2009; he came on as a substitute for Niall McGinn in the 67th minute, and forced a fingertip save from Scotland goalkeeper David Marshall with a 30-yard free-kick three minutes later. His under-21 debut followed three months later, when he came on as a substitute for Corry Evans in a 2–1 away defeat to Portugal on 12 August 2009. Norwood received national media attention in November 2009 after he scored free-kicks against both Germany and the Czech Republic's under-21 teams.
Norwood's reverse progression through the Northern Irish national teams was completed in May 2010, when he appeared for the under-19 side in the 2010 UEFA Under-19 Championship elite qualifying round. In his first match for the under-19s, against Russia, he scored two goals in the last 10 minutes to salvage a draw after Northern Ireland went 2–0 down shortly into the second half. He then followed this up with a penalty to open the scoring against Italy two days later, only for Italy to win the match 3–2.
Norwood made his debut for the Northern Ireland senior side on 11 August 2010, in a 2–0 friendly defeat away to Montenegro. He came on as a second-half substitute for Sammy Clingan, and in the final few minutes attempted to chip the Montenegrin goalkeeper Mladen Božović, only to send the ball over the bar. He started for the first time in the senior team when Northern Ireland lost 3–0 to Italy on their last game in UEFA Euro 2012 qualifiers.As of match played 2 January 2017
As of match played 15 November 2016.