Bursik spent almost his entire youth career at AFC Wimbledon before moving to Stoke City in 2017. Bursik was playing on loan for several other clubs, such as Telford United, Accrington, Doncaster Rovers, Peterborough and Lincoln City. At the beginning of the 2021/22 season, Bursik seized the opportunity and became the first-choice goalkeeper for Stoke City, a team that is active in the English Championship since 2018.
On 14 January 2023, Blauw-Zwart (Club Brugge) secured the services of the English youth international. The British player signed a contract until mid-2027.