One factor seldom mentioned that I find critical to both bridge battles and siege assaults: the squeeze factor. Whenever you crowd men together, so that they don't have sufficient room to wield their weapons properly, they'll attack and defend at significantly reduced effect.
The way around this is to set your unit's formation, before allowing it to engage, to a width that doesn't exceed the bridge or gatehouse width. On an assault across a bridge with the typical infantry unit, I find that a frontage of no more than seven columns (resulting in very deep rank formations) is ideal. I prefer polearms or swords to spear units, as their attack factors tend to be higher. Set to this frontage, I seldom (on Expert) need more than one unit to breach the enemy's frontage, unless they're massively equipped with missle troops. But in melee my men are fighting to full effect, while all the enemy's troops are struggling at half effect.
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