Born in Kingston upon Thames, Norman joined Fulham at the age of 9, turning professional in October 2014. He spent loan spells at Farnborough and Hayes & Yeading United. Norman joined Southport in September 2016 for an initial one-month loan. In October 2016 he was described as a "key player" for the club and in November 2016 the loan deal was extended until the end of the season. He returned to Fulham in March 2017, having made 29 appearances for Southport. He moved on loan to Rochdale in August 2018, and later joined Carlisle United on a 2 year deal in August 2020. On 1 July 2022, Norman joined recently relegated National League club Oldham Athletic on a one-year deal following his release from Carlisle.