He 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. 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. By the end of December, Norburn return to Leicester City after his loan-spell at Bristol Rovers finish and spend the rest of the season staying at Leicester City. On 3 July Norburn signed for 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 and 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, which he signed a one-year contract extension on 24 June 2013. In 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. He signed a new contract with Shrewsbury Town in July 2019.
