Instead of bags of ice, I just take some water bottles and stick them in the freezer. They keep everything cold, and then we drink them when they melt.
Most of the time, Dad drove. We had six kids, two of whom took turns in the front. Four in the back with Mom (or sleeping Dad). We lived out of the ice chest. The top surface was handy for Mom to make the sandwiches which she passed around while we flew down the road. We always had frozen jugs when the trip started and when we were ready to drive back. In between, the water got drunk and the rest of the chest needed to be replenished with purchased ice. We thought that was the way everybody traveled when we were kids. As a result of Dad and Mom's diligence, we ended up getting better acquainted with our granparents than some of the cousins who lived close, visited more often but for much shorter times.
Tried it a couple of times on my own kids. They were OK with it one way, but usually wanted Arby's, Wendy's or whatever on the way back.