Norburn played as a youth team player with
Chelsea before moving to
Leicester City, where he graduated through to the professional ranks in 2011. He joined
Bristol Rovers on a month-long loan deal in late September 2011, later extended to three months, where he played alongside fellow on-loan Leicester teammate
Cian Bolger. By the end of December, Norburn returned to Leicester City after his loan-spell at Bristol Rovers finished and spent the rest of the season staying at Leicester City. On 3 July Norburn signed with the Bristol Rovers on a permanent basis, and agreed to sign a one-year deal with the option for a further 12 months with the club. On the opening game of the season, Norburn made his debut in a 2–0 loss against
Oxford United. Two months later, he scored his first goal in a 3–1 win over
Northampton Town. His second goal came on 5 February 2013, in a 1–1 draw against
Cheltenham Town and his third was followed up a month later, in a 2–1 win over
Exeter City. At the end of the 2012–13 season, Norburn was offered a new contract by the club, where he signed a one-year contract extension on 24 June 2013. In the 2013–14 season, Norburn had his first team opportunities limited, making sixteen appearances. On 4 April 2014, he left Bristol Rovers by mutual consent. First team manager
Darrell Clarke said of Norburn leaving: "Ollie has left the club, and that was mutually agreed. For whatever reason it has not gone well for Ollie this season. Ollie has a chance now to try and earn himself a contract at another club, and we wish him well and thank him for his efforts." On 23 July 2014, Norburn signed for
Plymouth Argyle as a free agent on a one-year deal. He was released by Plymouth at the end of the 2014–15 season having failed to establish himself in the first team. He started training with
Guiseley in late 2015, and then signed for the club after impressing manager
Mark Bower in the sessions. On 27 February 2016 Norburn scored a controversial goal for Guiseley in a
National League match against
Braintree Town, after Braintree Town had put the ball out for a player to receive treatment. The goal stood and Guiseley drew the game 1–1. Norburn joined
Macclesfield Town of the same league in July 2016. He scored eight goals in his sole season with the Silkmen, including one in each leg of their
FA Trophy semi-final win over
Tranmere Rovers and one more in the 3–2 loss to
York City in the
final, before signing a three-year contract with Tranmere. 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. He signed a new contract with Shrewsbury in July 2019. On 10 August 2021, Norburn joined newly promoted
EFL Championship club
Peterborough United on a three-year deal for an undisclosed fee. On 14 December 2021, Norburn was made captain of Peterborough United replacing
Mark Beevers. On 29 June 2023, Norburn joined
EFL League One club
Blackpool. In January 2025, Norburn joined fellow League One side
Wigan Athletic on loan for the remainder of the season. On 16 June 2025, Norburn agreed to join League Two side
Notts County on a one-year deal with the option for a further season upon his release from Blackpool. ==International career==