I haven't slept for over 4 years 😂 welcome to motherhood!

I think it'd be abnormal if you HAD slept more than 3 hours at this point! My son will be 7 weeks tomorrow, and we ALMOST got  4 hours one time.  

If you're getting three hour intervals as far as sleep goes, you are in good condition.  My daughter is waking every 1.5-2 hours to be changed or fed (she is 15 days old).  Three hours is a good start.  

Have you tried using white noise? There are a few good phone apps that play white noise sounds to help calm your little one. When we were in the hospital my mother in law gave the baby a wind-up stuffed lamb. It was the only thing that got her to fall asleep.... But we had to wind it up every 3 minutes. We then found an app that plays all night. The nurses that came in at night loved the idea.