After Byrt took part in FIS Cup competitions since 2007, he was used for the first time in the Continental Cup on July 30, 2010 in Courchevel . On September 26, 2010, he managed to jump into the top three of a Continental Cup competition for the first time in Almaty, Kazakhstan . After further good placements in the Continental Cup, he was allowed to compete for the first time in the World Cup on January 21, 2011 in Zakopane as part of the national group . Two days later he won his first World Cup points with a 29th place. On March 12, 2011, he was used in the team competition in Lahti, Finland in the Polish team and achieved third place, the only World Cup podium of his career.
At the Polish Championships 2011 Byrt took second place on the normal hill and with the team of KS Wisła Ustronianka. On March 12, 2011, he and the Polish team took 3rd place at the Team World Cup in Lahti . On December 18, 2011, he achieved his best result so far in this competition with second place in the Continentelcup jumping competition in Erzurum, Turkey . In the same place he was able to win the silver medal at the Junior World Championships in 2012 with the Polish team, which also included Klemens Murańka , Bartłomiej Kłusek and Aleksander Zniszczoł .
After he was only used in the Continental Cup jumping competitions in January 2013 in Zakopane in the 2012/13 season and was disqualified for both competitions, he made another international appearance in February 2013 in two competitions in the FIS Cup at the same location , and proved there but only places 52 and 55. He then started until his end of career in 2016, apart from one start each in an FIS Cup competition in Štrbské Pleso and a Continental Cup competition in Brotterode, only in competitions in Poland.