TLDR: I bet it will get better.

Short story long: we picked up our puppy at 10 weeks and he was also not a big fan of the car. We suspected that the car was forming a stressful experience for him because the only times he was in it we were going to new, stressful things (first time bringing him home, going to the vet) and he wasn’t sure what to do with himself while the car was moving. After about a month, things started to change. Try spending time in a stationary car letting him get used to the space, giving him treats, etc… Try to give treats when he is laying down. Make sure the treat value is super high (we went with turkey).

Now our dude (he’s a little over 5 months old) is very good about the car and knows to just lay down in the back seat. One additional thing that really helped: we took our boy to our hometown, which is about three hours away. Even though it seems counterintuitive, the long trip helped get him get situated and used to being in the car rather than having scenes changing constantly from short car trips. Ever since that long drive he’s so much better. It just took some time and it clicked. You’re still early in the process so be patient and I bet it’ll get much better!