My 2nd daughter did this from about 2-6 weeks. It was so frustrating. I tried everything. It turned out that what she WANTED was to sleep on her tummy. I knew this because I tried, while I was standing right there, to see if she would fall asleep if she was on her tummy, and sure enough, she went right to sleep. On her back, she would fuss and wiggle and cry.
Of course, I wasn't going to let my baby sleep on her tummy. Any way that we can reduce the risk of SIDS is worth it to me. What I DID do was place a very, very soft blankie next to her face. She immediately turned her face towards it and slept with her face buried in the blanket. I wasn't exactly comfortable with that arrangement, but we all have our preferences when we sleep. She still, to this day, sleeps with her face buried in that same blanket.
Once I figured out what she wanted, she slept, from 6 weeks old, 12 hours at night. She still does.
Hope you find out what your little one is wanting! I know it can be frustrating! Hang in there, momma, this will pass!
ETA: I NEVER co-slept, never will, and always discourage other Moms never to co-sleep. Once you get them in that habit, it's so hard to break them of it. Some Moms don't mind, but I value my space & privacy (so does my husband), so I was never interested in that.