I have two allied city-states that are at war with each other. How can I stop this?
This isn't much of an answer, but I feel like it adds enough to be more than a comment. I've done a little research and it appears this case happens exclusively when each city state is allies with different civilization that are at war, thus at war with themselves. When the two city-states are allied to you they continue their wars.
Unfortunately, there is no way to force a city state to make peace with any other entity. Although, it is very rare for a city-state to actually conquer another city.
Actually, I think it IS possible (or at least that's how I did this) but you must be an ally of both city-states. Then you can declare war on one of them → the other one will declare war on him as your ally (which isn't too terrible given the already ongoing war between them ;D). Then you can immediately end the war with the city-state (by declaring peace, even in the same turn!) → this will make your allied city-states also make peace with that city-state (even if their war had been going on before).
It is worth mentioning, that in a city-state that you have declared war on - you will lose all your influence (it will drop to -60, as always when you declare war with a city-state). It is not so terrible if you can easily rebuild it with gold or, for example, freedom ideology skills.
*I'm not sure if it works with only one allied city-state, but I think it should.