Born in Rødovre, Bjur started playing football with Vanløse IF in the second-tier Danish 2nd Division. When the structure of Danish football was altered in 1991, Vanløse was placed in the second-tier Danish 1st Division. At Vanløse, Bjur showed himself as a forward with good finishing and eye for the game, and was named 1991 Danish 1st Division "Player of the Year". This attracted the attention of defending Danish champions Brøndby IF, and Bjur moved to the top-flight Danish Superliga club in July 1991.