I'm an level 65 archer with 93 archery and still need at least 6 headshots to take out a bear.
I use archery myself, slightly higher archery skill, a bit lower overall level (59) but there no way it should take 6 headshots to down a bear, I play on Master but it doesn't take that many unless you're using a very weak bow. Somethings not right somewhere.
What type of bow are you using and what damage does it do? You have improved it with smithing skills yes? What arrows are you using? Falmer and Forsworn arrows only do 7 damage, Daedric does 21. My Zephyr bow is smithed up to Legendary and I cleared out a Forsworn redoubt the other day without much trouble. I also have a daedric bow with fiery soul trap and paralysis, the last being particular helpful against massed enemy attacks.
I will admit that when I first started playing I was completely clueless as to how to do things and kept dying a ridiculous amount of times until I figured out the best strategies to use both in combat and to increase weapon damage and also to improve armour protection, enchantments are particularly useful in this regard.
So tell us, what armour are you using and whats its protection stats, what weapons do you use and their damage stats and finally what level of difficulty are you playing at?