My children are now teenagers and we still occasionally have this. Here are several ways to solve:
Settle specifics – In our house it has always been the music. Settling specifics means coming up with a full, solid and publicly agreed upon plan for repeat conflicts. For a few years our plan was structured around whoever was in the front seat picked first and on commercial breaks, control alternated. At some point a great debate started about what constituted a commercial break so we restructured. You might have a simpler plan like following odd/even days. On odd days one child makes all those decisions and on even days the other child.
Bring supplies and BOOKS – Stock the car with things to keep them busy. This might be magazines, notebooks and pens, magnet games, car bingo. Once children are reading, car rides provide an opportunity for them to really get into their stories. I remember reading whole novels on our trips to and from the grandparents house each year.
Give them something to listen to – Their music is a great place to start. Books on tape can be a helpful way to engage them. You might find these using the Audible app or at your public library. Listening with individual earphones might cut down on the bickering. Giving them just noise reduction earphones or earplugs while they color, play with their magnets or read and it might also reduce the bickering.
Sing alongs – Car rides are a perfect time for sing alongs. This might be to your children’s favorite CDs or teach them songs you know. I lean towards campfire songs and patriotic songs. Sadly, your children might not learn them otherwise.
Conversation starters – There are several companies that make question boxes. This includes Melissa and Doug, Table Talk and American Girl’s box of questions. These are a great way to start conversations that encourage everyone to participate.
Give them elbow room – It may be helpful to seat them farther apart. If you have a third row, consider moving one of them back there. Once the oldest is 13 years old, they might move to the front seat.
Put up dividers – When all else fails, divide and conquer. For about 6 months when I was in elementary school, my dad set a huge cooler in between us in the back seat. I couldn’t even see my brother let alone bicker with him. Cardboard might be easier.
Stop the car – When all else fails, fine to pull the car over and wait. Doesn’t help any if stopping the car is an empty threat. You’ve got to really pull over and wait them out. All the better if you are headed somewhere for them. My dad did this right before the cooler.