[3] In March 2016, whilst at FCB Escola Varsovia, Bułka underwent a trial with Barcelona, before trialling at English club Chelsea the following month.

[7] Bułka made his first and only appearance for PSG in the 2020–21 season in the first match of the campaign on 10 September 2020, which was against newly-promoted Lens.

With Keylor Navas and Sergio Rico absent, he would play the entirety of the match, but made a costly mistake that handed Lens a 1–0 win.

[citation needed] On 31 January 2021, Ligue 2 club Châteauroux signed Bułka on loan until the end of the season.

[20] On 7 June 2024, he was selected in the 26-man squad for UEFA Euro 2024 as a third-choice goalkeeper behind Wojciech Szczęsny and Łukasz Skorupski,[21] and remained on the bench for all three of Poland's fixtures at the tournament.