Bridge battles should mostly be fought if you're defending...
If you are on the offensive like it seems here, they are much harder as you have to either lure the enemy to your side or cross and bring the fight to their side.
The best way to defend in a bridge battle is to create a "pocket" where the AI can disgorge its troops but cannot deploy effectively.
This is done by placing your infantry (spears or swords) in a U-shape around the base of the bridge on your side.
Placing your archers on both sides of that U so that they can shoot on the massed enemy on the bridge.
Keeping your cav at the back to reinforce any breach in the U
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