Shouts at first arent that good IMO, have you tried getting Conjuration up? Summon a monster to fight for you (or reanimate a corpse) and they take all attention while you rain the arrows down on them. Also always poisen your arrows helps so much!
I am probably way behind the curve of everyone else, but I'm struggling occasionaly with higher powered mages. I'm playing a 'sneak with bow' style and can take most single enemies down with one or maybe two shots. (Level 17 Wood Elf, Sneak~50, Archery~40) But the higher levels mages tend to be a bit tricky with their shield/protection spells and fireballs.
How do others cope?
Could it be because I haven't even got my first dragon soul yet so can't use any shouts...
You have an imagination barrier.