Hiya I had this problem with both of my twins.
We have one of them on infant gaviscon which has really helped, it didn't suit the other one.
They threw up after every feed and when going to sleep. They always slept on their backs and never chocked, even though they throw up.
After speaking to HV and midwives they just said as long as they put on weight and are generally happy, then it's fine, just carry on and buy lots of bins and muslin cloths!
My DS had to go on gaviscon after a week he was throwing up through his nose and screaming like mad, gaviscon changed him into a happy baby.
It didn't agree with DD but she was never as bad.
We had to wonder how they were putting on so much weight and throwing up so much! But they are completely fine! It wasn't due to over feeding either!
Try not to stres about it, as long as they are happy and putting on weight. They won't choke on their backs, they can turn their head to the side that way.
Apparently it gets better when they are on solids!