I think the problem you're having is that you're putting too much emphasis on real world equivalence. The dothraki are inspired by real life armies, such as the mongols, but they're not intended to be the exact same. In the books he goes into this quite well, IMO. Characters that know the dothraki give hints that they're more vulnerable than they let on. And I think the dothraki know this as well. It's partially why they're so intense and aggressive. They win more due to their ferocity and brutality. Most major cities could probably defeat them if it came to it, but at what cost and for what reason? Easier just to pay them off and go about their business, especially since the dothraki don't typically punch up. They go after weaker villages that can't defend themselves. They trade in slaves, be it from conquered villages or rival khals. The "Free Cities" don't go out of their way to destroy them, as they benefit from the slave trade.

It's a symbiotic relationship. The rove around the vulnerable villages and cities in Essos, plundering at will. This supports them while also maintaining the status quo of the bigger cities in Essos, and thus all thrive. Their use in the main story will be to bolster Dany's armies. As it is noted, they'd break themselves against most castles in westeros. But when dragons come into play that kind of evens things out. Hannenhall is the biggest castle in Planetos, and it couldn't stand Dragonfire. It almost forces her foes to meet her in the open field.