My baby won't sleep in her crib!

Ever since my daughter was born I've noticed she hates sleeping completely flat on her back. She's been sleeping in a bassinet that is slightly elevated since then. Originally this bassinet was only supposed to be used for travel since her dads family lives in a different state but it's become her everyday sleeping arrangement. She's a little over a month old and I've been trying to get her to sleep in her crib now and nothing is working. I make sure she's always content and in a DEEEEEEP sleep before I put her in there and as soon and she's realizes she's on her back she starts fussing. Do you guys think as she gets older this will go away? My plans now are to maybe just let her sleep in this bassinet until she's around 3 months and try again. However, I'm nervous that this wont work because I was hoping but a month she could sleep in it & that obviously isn't working lol. Any advice?