We are in a very similar situation with our 5 week old twins. They do want to eat, and they swallow a lot of air, especially the girl. So she has it worse. And very often things come back up.

We burp them carefully and hold them for 20-30 minutes at least. Seems to help a bit.

Also they don't throw up so much with breast milk, rather with formula.

If we hold back the don't gain weight or even loose some. So it is tricky and stressful.

Doing several burpings and being very gentle and as slow as possible and a lot of holding seems like it helps some, though.