Born in Eastbourne, East Sussex,[2] Norwood played for Brighton & Hove Albion and Crystal Palace youth teams.

[7] Later in that same season, he played for Eastbourne Town's senior side despite still being at school, making eighteen appearances for the Isthmian League Division One South club, scoring ten goals.

[12] Norwood made his debut in Exeter City's 2–1 defeat away to Leeds United in League One on 8 August, the opening day of the 2009–10 season.

[13][14] Exeter manager Paul Tisdale commented on his debut saying; "It's remarkable that James should be on the field after only joining us a few weeks ago.

"[13] Norwood joined Isthmian League Premier Division club Sutton United on a one-month loan deal starting on 21 November 2009, which was extended into January 2010 for another month.

[22] Norwood returned from his injury as a second-half substitute on 26 November 2011 against York City and scored a last minute equaliser to earn Forest Green a draw.

[24] On 11 February 2013, Norwood signed a new two-year extension to his contract at Forest Green, a move that would see him tied down to the club for up to the summer of 2015.

[31] He helped Forest Green to the Conference Premier play-offs for the first time in the club's history, although they were knocked out in the semi-finals in May 2015 by Bristol Rovers.

[34][35] Norwood's opening weeks with Tranmere were dogged by injury issues but he finally found the net for the first time against rivals Chester in mid-September.

[36] Norwood had netted nine times for Rovers come the turn of the year, including a looping header in a 2–0 victory against former club Forest Green and Tranmere's eventual goal of the season in a 4–1 away defeat at Woking.

[41] Norwood finished the 2016–17 season strongly with 5 goals in as many games including one in each leg of the play-off semi final against Aldershot.

[45][46] Two of these goals came in the play-off semi-final versus Ebbsfleet, the first being a clinical left footed finish into the far-bottom corner and the second being a curling 25-yard extra time free-kick that put Tranmere 3–2 up.

[47][48] On 12 May 2018, almost three years to the day of his arrival at Tranmere, Norwood scored the winning goal in the play-off final against Boreham Wood at Wembley to ensure the club's return to the EFL after a three-year absence.

[51] Norwood joined Ipswich Town on 10 June 2019 on a free transfer, signing a three-year contract, with the option for a further year.

[58] On 1 October 2020, Ipswich announced that Norwood had suffered a torn hamstring in training and that he would be set for a spell on the sidelines, with the injury expected to keep him out of action for at least six weeks.

[59] Norwood made his return from injury earlier that expected, coming on as a second-half substitute and scoring in a 3–2 loss to Portsmouth in an FA Cup tie on 7 November.

[70] Norwood experienced his first taste of representing his country in March 2009 for the England Schools squad in a Carnegie Centenary Shield fixture against Wales.

[72] He scored a last minute penalty on 19 November 2013 in a friendly with the Czech Republic under 21s squad to earn England a 2–2 draw at Woking's Kingfield.

[74] On 4 February 2015, alongside teammate Elliott Frear, he was called up to the England C squad for a friendly with the Cyprus under 21 set-up later that month.