I'm not sure whether this will be accurate or not. To my opinion, 'walk on the bridge' means walking on it itself while 'walk over the bridge' is like walking across the bridge, from this end of the bridge to the other end of the bridge.. Whenever people ask u direction, you may say, walk over the bridge and u will reach the place. I hope it helps.

"Walk on the bridge" emphasis more on the action of being on the bridge. "Walk over the bridge" emphasis more on the location on the other side of the bridge.