Just as with everything else, practice makes perfect. When you rise your thieves guild level, passives are unlocked which reduce the time a bounty sticks and which reduce the sight range of guards.
Regarding the fighting; a thief wants to move in, steal the goodies and move out without being seen. Not being seen means there will be no bounty. There are hiding spots in the trespassing areas. With a bit of patience and observing the movement of the NPC's present, you can sneak and steal unseen. Even in case of being seen, you can go in a hiding spot and wait until the bounty has dropped before trying to sneak out again. (Yes, the guards will stop chasing you if you go into hiding, even if you jump into a big basket in plain sight of the guards).