Bugaj started playing football with the youth side of Molde FK. In 2019, Bugaj progressed to the Molde FK 2 team, and made his league debut in a match against Eidsvold TF.[2] Despite the promise shown by Bugaj, he failed to make another appearance for the second team. In 2020, Bugaj had a trial with Borussia Dortmund,[3] however no offer was made by Borussia and he joined SPAL on a permanent contract.[4]
Upon joining SPAL, Bugaj was placed in the club's academy and played with the youth teams for the next three seasons. On 31 July 2023, he joined Polish second-tier club Lechia Gdańsk on loan for the 2023–24 season.[5] Throughout the season, he helped the club win the league and gain automatic promotion back to the Ekstraklasa.[6][7]
On 16 July 2024, Bugaj moved to Major League Soccer club CF Montréal on a two-and-a-half-year deal, with an option for another two years.[8]
His father, Radosław, was also a footballer. He spent 10 years playing in Poland, most notably with Zagłębie Sosnowiec where he played in the II liga, Poland's second division at the time.[9] In 2007 when Dawid was 3, his father moved to Norway, settling down in the country after the conclusion of his playing career. His younger brother, Alex, is also a footballer, currently playing with Wisła Kraków II.
