Yes, normal indeed. I had trouble sleeping the first couple weeks after quitting drinking but eventually got used to being sober and started sleeping normally. I now sleep like a baby and wake up feeling good.
It does take a while before your body/brain adjusts to being sober, so give it time. If your sleep problems persist too long, see your doctor. Besides, passing out from drinking is not normal or restful sleep, so it's really not 'sleeping' at all, more like just being unconcious.