Sleep Problems

So my baby is 2 months old, and he will not sleep on his back. He sleeps in his swing, I know it’s not recommended but that’s the only way he sleeps. He has reflux really bad and he will spit up a lot if he’s on his back after a bottle, even 2 hours later. I don’t understand it. When he was less than a month old he did not have spit up problems but he still did not want to sleep on his back, he hated the bassinet. My question is, how do I transition him from the swing to the crib? He’s trying to roll so I don’t want him sleeping in his swing anymore. Any help is appreciated