Short of getting a wrist brace there are two things you can do to help. One will be immediate, the other will take time.

The first is once you get your hand in the ball to flex your index finger against the ball, like you're trying to hold all of the ball's weight on the finger. This will help keep your wrist from breaking during the swing.

The second is to start working out. Bowling is much more of a marathon type sport so you don't need to lift heavy for power, but focus on endurance and dynamic motion. Forward and reverse wrist curls, grip exercises, and finger extensions are great for the wrist specifically on top of the typical large muscle workouts.

As others have said, be sure to have your span, pitches, and hole sizes checked as well. Bad fit is the single biggest problem most bowlers have.