Biegun made his first international appearance at the age of 14 at a junior competition in Hinterzarten . Two and a half years later he was used for the first time in the FIS Cup in Štrbské Pleso, Slovakia, in January 2011 , but only finished 36th. He first attracted international attention on February 17, 2011 when he played with the Polish youth at the European Youth Olympic Winter Festival in Liberec, Czech Republic Equipe was able to win the team competition. In the individual competition he had only finished 26th two days earlier. Exactly one year after this success he made his debut in the Continental Cup in Oslo , where he finished 52nd. Since then he has mostly jumped in this second-rate competition series.
On July 21, 2012 Biegun was used for the first time in the Summer Grand Prix when jumping in Wisła as part of the national group and was able to collect his first points at a higher level as 20th. Then a month later he was used in the Japanese Hakuba in the Grand Prix, but could not collect any further points for the overall standings. Also in Wisła he won his first medal at national championships with second place behind Maciej Kot at the Polish championships on the large hill.
In the Junior World Championships in 2013 , which like the European Youth Olympic Festival two years previously also held in Liberec, he could on 26 January 2013 with the Polish team to the next to him Bartłomiej Klusek , Aleksander Zniszczoł and Klemens Murańka were behind Slovenia win the silver medal. After he made it into the points three times at the Continental Cup competitions in Iron Mountain in the US at the beginning of February, including his best result in this competition series with seventh place, Biegun was the first weekend for the Polish World Cup team nominated: When flying from the Heini-Klopfer ski jumping hill , he not only reached his first World Cup point in 30th place, but also made the longest flight of his career with 180 meters. On February 23, 2013, he was able to climb the podium of a Continental Cup for the first time in Wisła as third. At the end of June 2013, Biegun won the first Continental Cup competition in Stams, Austria . On August 23, 2013 he won an individual competition in the Summer Grand Prix in Hakuba, Japan , after winning the team competition in Wisła with the Polish team three weeks earlier . In the overall ranking of the Grand Prix , he finished fifth. His first World Cup victory followed on November 24 of the same year in the Vogtland Arena in Klingenthal . At the Junior World Championships in 2014 in Predazzo , Italy , he was team world champion together with Zniszczoł, Murańka and Jakub Wolny .
Subsequently, his performance curve showed down. He was able to win a Continental Cup competition for the second time on February 7, 2015 on the Inselbergschanze in Brotterode and three weeks later on the Hochfirstschanze in Titisee-Neustadt again, but in the World Cup and in the Summer Grand Prix He was denied further successes. After another third place on December 17th, 2016 on the Rukatunturi hill in Kuusamo , Finland , no more countable results followed. In October 2018, Biegun announced his retirement at the age of only 24 and justified this step with disappointing results and the resulting lack of motivation and weight problems.