Still trying out bridge crossing suggestions.
One way that works (if you're rubbish like me) is too build up a huge force, much bigger than the enemies and then let the computer auto-calc. The computer seems to work out the battles on size of army.
I do it like this coz I'm still really bad at crossing bridges.

but....

It seems to me that early in the game when you only have SA and YS it all comes down to attrition. Throw your men across the bridge in an effort to draw the enemy forward into your archers range, if the enemy pursues your routing units so much the better. (I always use Ashigau for this bit, as it doesn't lower my mens moral when they leg it.)
Then its in with your YS who usually rout.

The only times I've won a bridge crossing is when I rushed my Generals unit in, my men don't rout (at least for a little longer) and hopefully (like Ii Naomasa said) I can knobble their General before my men go chicken, and my own General gets slaughtered.