There are a few things that you must have in a bridge battle, siege weapons, archers, the will to kill units. It pretty much allows you to defend and attack. After your siege runs out of ammo, send them to die, when your archers run out of ammo, send them to die. Remember to always have atleast 2 archer units in reserve. Use one unit to draw enemy fire, you know, waste their ammo, but save yours if the enemy goes out of range. Have 3 elite infantry units, and plenty of cav, especially a few light ones.
When you're attacking, you can use a light cav, especially on those barbarians, to draw enemy untis into chasing, then the kill begins. When you feel that the enemy is weak enough, destroy them.
WHen you're defending, don't position yourself close to the bridge, let the enemy cross, your archers will kill them while they do. BEsides they still have to line up after they cross
well those are my 2 cents.
BTW bridge battles are easier on rome, but harder on medieval and shogun, especially if you play multi.