Bednarek signed with Dutch side FC Twente at the age of 16. He made his debut for FC Twente on 6 December 2012 against Helsingborgs IF in UEFA Europa League.[1] As he never could make it to the first eleven, he left Twente as a free agent in July 2015. After being on trial with FC Utrecht, he signed a one-year contract with the club, including an option for an extra season.
He returned to Poland on 5 June 2020, signing a two-year deal plus a one-year option with Ekstraklasa side Lech Poznań.[2] With Lech's then-captain Mickey van der Hart out with injury, Bednarek was Lech's first-choice goalkeeper throughout the first half of the season, including their entire Europa League campaign.