Under Brøndby manager Morten Olsen, Bjur saw a lot of playing time, but had a hard time scoring goals. When Brøndby hired new manager Ebbe Skovdahl in 1992, he went on to deploy Bjur in a right wingback position, where Bjur was immediately praised for his performance. From his wingback position, he would both engage in elegant, dribbling runs as well as hit precise crosses. In the opening of the 1993–94 Superliga season, Bjur saw a slump in form. He got his only red card while at Brøndby in September 1993, and was eventually left out of the Brøndby first team and played a number of games in the reserve team. He returned to form in October 1993, and became one of Brøndby's leading men for set pieces. With Skovdahls change of tactics, to a 3–4–3 formation in March 1994, Bjur was moved up to a forward position, behind the lone striker Mark Strudal, where he reproduced the attacking play of his Vanløse days.