I always get to a point with all my thief characters where they inevitably get into a battle and have no chance with low armor rating and weak weapon skills. They die a terrible death. That is the point when I abandon the character.

What can I do to remedy this?

Take a (sneaky) follower with you?

Well, there are two things you can do. If you play only a thief, as in, not a jack of all trades, or a warrior thief, or something, just don't get yourself in situations where you have to fight. If you do plan to fight then you need to invest some points into health, and perk your light armor and one handed skills (or whatever else you decide to use).

Of course, your problem might be that you simply level up too much by using exclusively thief-related skills and by the time you get in a fight your lv 15 armor skill and offensive skill can't support you. Here's one way you can remedy that. Anyone who played any kind of a thief in Skyrim knows that thieves are the richest characters in the game. Simply do what you do from level 1, and after each level increase try to spend your "hard earned" septims for training in armor and offensive skills of your choice and you should be fine.

I think it depends on what "other" skills you use, besides thief skills. Do you use Illusion magic? Archery? Destruction magic? All of these can help keep battlers at a distance or help avoid them all together. Are you playing as a Dragonborn? Unrelenting Force is an excellent way to make opponents "back off" and give you a breather.

Just some suggestions to go with Chaka ZG's above mine

"He who gets caught and runs away..."

is undead! Kill it! Kill it with fire!

Yes to all of these.

I suggest you find a sneaky follower until you can get your sneak skills maxed. I always use the buddy system in the beginning, especially one that can soak up the damage while you either work on your one handed skill, run away, or hide until your follower deals with them. Barring that I'd suggest you do your best to avoid combat altogether. It's rough, but can be done. I have a Khajiit thief that doesn't fight or even carry a weapon. I had to work a long time on just his sneak skill, because he doesn't ever take a follower.

My Khajiit also doesn't use magic or stamina for fighting, so I dump lots of points into his health. I found a 4:1 split feels just about right for me, because you still need "some" stamina to sprint away. He does a lot of sprinting, even now.

Even still, I would counsel you to consider that open combat is an inevitability. There are going to be points which you can't sneak past enemies unseen, and so investing a lot of time and effort in your alchemy skill to come up with some powerful paralysis poisons to coat arrow tips may be your best option. Assuming you choose to abide by the Thieves Guild rule of no killing.

All my thieves are archers. I usually level up hunting using the sneak shot damage which levels up both abilities (but I'm sure you already know it) Then I have a strong offense. Also is a good idea to have a powerful Conjurer with you since they distract more the enemy, allowing you to shot or even get hidden and shot.