There is an easy and FAA-approved child restraint system called CARES. My wife and I purchased one and used it with great success. The flight attendants had never seen one, so it comes with a card for you to show them.
Notably, though, German carriers like Lufthansa do not allow it, which I believe is due to German regulations. Outside of that, we've never had an issue in either the U.S. or Europe, but occasionally we need to show the airline a copy of their policy from their website on one of our phones. They weren't mean about it, but may not have seen one. Generally, the senior flight attendant is experienced enough to know the rules.
Our kids were huge, so we used it from before they were a year old and it has been very helpful.