Before joining Podbeskidzie, Byrtek spent time with Olimpia Piekary Slaskie.
In February 2010, he signed five-year contract with senior team of Podbeskidzie Bielsko-Biała.
In January 2016, Byrtek moved to Wisła Płock. The club were promoted to the Ekstraklasa in his debut season, and his top-flight debut came on 8 August 2016, in a 2–2 draw with Ruch Chorzów. He was brought on for Tomislav Božić in the 85th minute.