Bistrović made his senior debut for Slaven Belupo on 30 May 2015, coming on for Petar Mišić in 77th minute as Slaven lost 3–2 to Osijek.[3][4] His proper debut came during 2016–17 season after the winter break, when he joined the senior team permanently.[3] In summer 2017, during the preseason, Slaven drew 1–1 with Russian championsSpartak Moscow in a friendly match in Villach on 7 July. Slaven's equalizer was a joint effort of Bistrović and Bruno Bogojević, who netted it in.[5] The match helped Bistrović attract attention of the Russian market, with Spartak's rivalsCSKA Moscow's coach Viktor Goncharenko insisting that the club buys the young Croatian.[3]
CSKA Moscow
On 12 January 2018, CSKA Moscow signed Bistrović for €500,000 until the summer of 2022.[6][7] At the time of signing for the Moscow club, Bistrović had made only sixteen appearances for Slaven.[3] On 8 March, he made his debut for the team in the UEFA Europa Leagueround of 16 match against Lyon where he came on as a substitute for Alan Dzagoev in the 82nd minute.[8] In the quarter-final home leg against Arsenal on 12 April, he was praised for his performance as CSKA drew 2–2, but nevertheless failed to progress to the semi-finals.[9]
On 20 September 2020, he scored the only goal in the 1–0 victory over Ufa.[19] Seven days later, he received two yellow cards and was sent off as Lokomotiv Moscow defeated CSKA 1–0.[20][21] On 10 December, he scored his first European goal, as CSKA lost 3–1 to Dinamo Zagreb in the Europa League.[22] It was notably the first and only goal Dinamo conceded in the group stage.[23]
Loan to Kasımpaşa
On 22 January 2021, he joined Kasımpaşa on loan until the end of the 2020–21 season. Kasımpaşa had an option to make transfer permanent at the end of the loan term.[24]
Loan to Fatih Karagümrük
On 12 January 2022, he joined Fatih Karagümrük on loan until the end of the 2021–22 season, with Fatih Karagümrük having an option to make transfer permanent at the end of the loan term.[25]
Loan to Lecce
On 24 July 2022, Bistrović joined Serie A club Lecce on a season-long loan, with the option for a permanent transfer.[26]
Loan to Fortuna Sittard
On 17 January 2023, Lecce announced that Bistrović agreed to a new loan to Fortuna Sittard in the Netherlands.[27] Fortuna confirmed the loan two days later.[28] He made his debut for the club on 25 January 2023, starting in a 2–0 league victory against SC Heerenveen.[29] On 3 March, he scored his first goal for Fortuna, helping the side to a 2–1 comeback win over Utrecht.[30]
Loan to Baltika Kaliningrad
On 14 September 2023, Bistrović moved to Russian Premier League club Baltika Kaliningrad on loan with an option to buy.[31] Baltika's manager Sergei Ignashevich was the teammate of Bistrović at CSKA in the 2017–18 season. In his debut game for Baltika on 16 September 2023 he scored an added-time equalizer in a 2–2 draw with Rostov.[32]