I learned a valuable lesson form my aunt who has 10 kids when I mentioned to her my first kept me up all night nursing for an ENTIRE YEAR.
Sleep training is largely never necessary because sleep is based on natural body clock and fullness. A full baby will not wake or need any elaborate sleep routines. Many times people don't realize their babies are hungry and could eat more. Cosleeping and nursing on cue, while not wrong, do train the child to comfort themselves by nursing and cosleeping. Babies who go right into a crib get used to that right away, so it's all a personal choice, but the trick to the baby SLEEPING in either case is having a full belly. The CHALLENGE is that nursing babies are never full!! They need to nurse a lot!! Which is why my first kept me up so many months..I exclusively nursed for as long as possible and fed on cue..as soon as I started adding feedings to her days and more bulk to her diet, she slept without trouble.
My second was a big, ravenous, baby boy, and to be honest, I could not nurse him enough for my lifestyle at the time. I was home alone with a 2 year old, and I needed sleep. So I added formula and cereal and food as early as possible on top of nursing and stuffed him. So he slept through the night by 3 months. My third was a girl who I also stuffed, and she slept through the night even earlier because she wasn't as big and hungry.
Anyway. Once you can feed him more, he'll be more satisfied and sleep through the night. Hang in there, it's not easy! And if he's nursing strictly for comfort a lot, don't feel guilty to move him away from you and let him learn to feel secure sleeping alone. It's perfectly healthy too if he's full enough.