During the 2018–19 Czech Extraliga season, Bednář made his professional debut for HC Karlovy Vary at the age of 16, becoming the youngest goaltender to play for Karlovy Vary. He recorded a 2.73 goals-against average (GAA) and .917 save percentage in 10 games. During the 2019–20 Czech Extraliga season, he recorded a 4.39 GAA and .884 save percentage in 13 games for Karlovy Vary. He also spent time with HC Banik Sokolov of Czech 2.liga, where he posted a 3.26 GAA and a .873 save percentage in 24 games.[1][2]
Having concluded his junior career with the Titan, on 15 March 2023, Bednář embarked on his North American professional career in signing a two-year AHL contract with the Grand Rapids Griffins, the primary affiliate to draft club, the Detroit Red Wings.[5] He was then assigned to ECHL affiliate, the Toledo Walleye for the remainder of the 2022–23 season. He appeared in three games with a 1–1–0 record, a 3.01 GAA and a .901 save percentage. During the 2023–24 season, in his first full pro season in North America with the Walleye, he appeared in 36 regular season games and posted a 22–7–2 record, with a 2.66 GAA and a .901 save percentage. On 3 June 2024, Bednar signed a one-year contract extension with the Griffins.[6]