Three things
1. Use your back muscles to support your position on the bike.
2. Grip the tank with your thighs - as if your life depended on it.
3. And relax.......
Your main problem I would suggest is that youre being a bit lazy with use of your back muscles and trying to support the top of your body with your arms, hands & wrists. This also makes you 'tighten up' and become un-relaxed on the bike. Use your back muscles and put very little weight on your bent arms - don't grip the handlebars (lightly caress them).
If you follow the above advice - your wrists will not hurt.
