in a bridge battle is is considerably better to put units at the end of the bridge (not on it). make a kind of hollow box walled by your units (the heaviest infantry you have (if your have spears then wonderful)). each unit they send across the bridge will be automatically flanked from both sides. stick all archers and such to both sides, as close to the water as you can (this way they can pepper the enemies sides as they cross the bridge). keep any cavalry in reserve, and if they every get close to breaking out of the box smack them with a cavalry charge.
added bonus: all of their units will get all clustered up in your box, so onc you route one, then you probably route two... which nearly always sets off a chainreaction of fleeingenemies. fear offsets any statistical disadvantages unavoidable if you meet them on the bridge itself.
the only downside is if they take a cautious skirmisher-heavy approach you'll take a lot of causualties due to your own packed formation