In the opening sequence, the point is not to avoid getting shot.
Instead, the point is to help the girl, whatever the cost. You're
going to have to be in close to do that.

While you're chasing the girl in the white cloak, you'll see various
hot spots in the woods. Whenever the hot spot appears on a log or a 
chunk of wood, go for it! That's the only way to get through the 
sequence, and to do it you have to fight off the king's men and/or
escape several times.