It refers to the kind of action he is making to move the bike, "pulling" the handlebars to turn it. It makes the sentence sound more action-packed than just saying "turned".
By "the street opposite" I would understand that he had to cut across the traffic, so overall the sentence is describing some reckless or hasty driving on the part of the cop.
I think you have cut a bit so it makes slightly less sense then it should do, but that's what I understand.