When we left hospital with both DS and DD, the hospital rule was that if you were leaving with a car seat, you had to bring the seat upstairs, put baby in and let a midwife inspect it before you went home. To make sure you had tightened it enough. With DS, we had almost done so but she recommended a little firmer. Clearly with DD there was no problem.
But I KEEP seeing people with new babies who don't. SIL kept banging on about how once he was in the actual car the seatbelt tightened (How? It's irrelevant) and some friends had their tiny newborn in without any tightening whatsoever the other day?
I am not usually a judgy person, but surely using a seatbelt correctly is kind of an obvious thing to check you know how to do before you put your baby in a few tonnes of metal moving at 70mph?!!?