Norburn played as a youth team player with Chelsea before moving to Leicester City, where he graduated through to the professional ranks in 2011.
[3] He joined Bristol Rovers on a month-long loan deal in late September 2011,[4] later extended to three months, where he played alongside fellow on-loan Leicester teammate Cian Bolger.
[3] He made his debut in senior football on 14 October 2011, when he was an 86th-minute substitute for Bristol Rovers in a 5–2 victory over Rotherham United.
First team manager Darrell Clarke said of Norburn leaving: "Ollie has left the club, and that was mutually agreed.
[15] On 9 August 2018, after winning promotion to League Two with Tranmere via the play-offs, he signed a three-year deal to move to Shrewsbury Town, led by his former Macclesfield manager John Askey.
[17] On 10 August 2021, Norburn joined newly promoted EFL Championship club Peterborough United on a three-year deal for an undisclosed fee.
[20] In January 2025, Norburn joined fellow League One side Wigan Athletic on loan for the remainder of the season.