For the 1994–95 Superliga season, Skovdahl changed the formation to a 4–4–2 system, and Bjur settled into the right-sided midfielder spot. He found a good understanding with right back Søren Colding, and the two formed a dangerous pairing on Brøndby's right flank. Bjur was a part of the Brøndby which won the 1994 Danish Cup, his first trophy in his Brøndby career, and was named 1994 Brøndby IF "Player of the Year". In the 1995–96 Superliga season, Bjur played all 32 games as Brøndby won their first Danish championship in five years. He also scored the winning goal in the 2–1 defeat of A.S. Roma in the 1995–96 UEFA Cup tournament, though Brøndby were eventually eliminated on the away goals rule.