I've lost several bridge battles, mainly though strategic incompetence (not enough troops / no decent general).
There was only one time it mattered. This was in STW, I was playing the Hojo and defending Musashi against the combined might of the Takeda and the Imagawa.
I had two units of archers (one of which was general), a unit of yari samurai (pike men) and a unit of yari ashigary (peasants with pikes).
Archers withered down my yari samurai as the enemy charged of the bridge. My troops put up a good fight, but the constant streams of arrows and enemies took its toll: my men began to waver. I had decided to keep the ashigaru back since I didn't reckon them to be able to stop the first wave. Now my samurai were wavering I thought that the support of other troops, any troops, would encourage them to fight on, so I plunged my ashigaru in. I thought wrong. The samurai routed and of course the peasants lost the will to fight when seeing that. I knew the battle was lost and withdrew my archers in good order.

End result: lost most of my spear troops, but none of my archers. Inflicted 1.5 or 2 times my casualties on the enemy. I faced a though fight next turn to liberate the province and rescue my heir, but I managed it.