In the 2018/19 season, Bury took three wins in the Slavic Cup and one second place and thus achieved second place in the overall standings at the end of the season. At the U23 World Championships in Lahti in 2019 , he came in 40th place in the sprint, 19th place over 15 km freestyle and 18th place in the 30 km mass start race. At the season highlight of the Nordic World Ski Championships 2019 in Seefeld in Tirol , he ran to 62nd place over 15 km classic, 47th place in the sprint and 18th place together with Maciej Staręga in the team sprint. At the beginning of the 2019/20 season, he scored his first World Cup points in the Ruka Triple, which he finished in 46th place, with 19th place in the pursuit. This was followed by 24th place in the Tour de Ski 2019/20 and victories at the Slavic Cup in Zakopane over 15 km classic and in Štrbské Pleso in the sprint. He also won the classic Bieg Piastów over 50 km and at the end of the season he finished sixth in the overall ranking of the Slavic Cup.