As others have mentioned he does have super strength but i would argue thats not the whole of the story. He is usually moving with the web not being tugged in random directions so the force would be similar to swinging on a rope swing, which many normal people do without breaking their wrists. The tugging would be more likely to dislocate or rip off rather than break. Most broken wrists occur from bad angles of the force or from landing on the wrist, not from a tugging or swinging motion.

My anecdote to contribute is i once had a rope wrapped around a solid object in a lake. The other end was loosely looped around my wrist while i was on a pontoon boat. The driver was backing up so we could do some measuring and i ran out of rope, it damm near pulled me off the boat but my wrist not only stopped the pontoon but kept it in place against the motor breifly without breaking. Its all about the angle of the force.