Can I exit the US with my French passport and come back in the US with my American passport? [duplicate]

US law allows this. It requires you to have a valid US passport when you leave the US, not to use it (the actual word used in the statute is "bear"). I do it all the time (with a Dutch passport). There are two possible problems, neither of which I have ever faced.

The first is that the airline might insist on seeing your US passport, or they might insist on seeing evidence that you entered the US using your French passport. I usually fly on European airlines, and they have always accepted my Dutch passport, but I have heard various things suggesting that US airlines might be less inclined to do so. If that happens, simply show both passports.

The second possible problem is that the country's immigration authorities might insist on seeing the passport that has been submitted by the airline in the advance passenger information record or any other passenger manifest. To prevent this from being a problem, if you have to show your US passport when you check in make it clear to the agent checking you in that you will be using the French passport to enter the destination country.

Finally, make certain before you travel that the restrictions imposed by your destination are indeed based on citizenship rather than residence or physical presence. You say "the country I am going to accepts French citizens," but if the restrictions are actually based on residence or presence then you might be unable to travel.