My Imperial faction character is having a lot of trouble holding back from putting an arrow between his shoulder blades.

I lol'd. Don't let the door hit your ass on your way out, Milk drinker.

Wow...what a thread, and what a post by the OP...it boggles my mind...shut up and play the game or don't play it...I'm tired of cleaning snot up off the floor from all the crybabys...

Bandits should not be able to kill you in two hits. Either you are exaggerating, or there is something else wrong. For starters, regular bandits do not scale that high. They become bandit outlaws, thugs, etc. Bandit leaders might be able to kill you in two hits if they're using two-handed power attacks and you're not bashing, blocking or dodging, or you're letting them hit you in the back, but those aren't regular bandits and they don't come in packs. Bandit marauders start appearing near your level, but even then, if you're taking perks in Heavy Armor you should be able to withstand more than two hits.

Some other information which might be useful:
- What difficulty are you playing on?
- What quest are you doing, specifically, and in what dungeon? The companions offer radiant quests - which are semi-random - in addition to their main quest line, which doesn't feature bandits. When you say "The second companion quest" we don't know which one that is in your game. Some dungeons are harder than others, and you may be attempting a quest which is above your level.
- Are you diseased? Some diseases can damage your max HP, obviously.
- Are you encumbered? This can make you unable to dodge effectively, and artificially increase the difficulty of an encounter.
- What is your armor rating?
- How often do you improve your armor? Improving your armor will usually provide a bigger damage reduction than finding a new type of armor would, and this is doubly true with the relevant armor perks. If you have been neglecting to improve your armor at the workbench, your damage reduction will almost certainly start to fall behind in the 14 levels between Steel Plate and Ebony. Merely upgrading to Orcish won't help much, and it will only get worse in the 8-16 levels afterwards if you don't improve them in the interim.

For perspective, I have a level 30+ character without any perks at all, and even when he was still wearing unimproved steel plate with ~30 heavy armor skill and suffering a -60 HP penalty from a particular disease, he could still endure more than the two hits you're reporting, at least on normal. The fact that your character sounds so fragile implies you have a very low damage reduction, unless these bandits are receiving a 500% damage bonus against you.

As you can see OP, badmouthing the game on this forum means everyone hates you.
Try a new approach...
Oh what a wonderful game I love the leveling system that Beth hath bestowed upon us.

Closed for review.

Threads started in anger and frustration tend to attract angry and frustrated replies, even moreso when the title of the thread seems calculated to put fans of the game on the defensive. That doesn't excuse some of the behavior exhibited in this thread- indeed, some of you will be hearing from me- but it does explain why this thread is staying closed.