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.
Could you ask another Civ to go to war with one of the city states and allow it to capture it, then declare war on that Civ and capture the city back and liberate it?