Roxy, I bet you lock your elbows for some reason. Check the reach/fit, strengthen your core, and really focus on being relaxed: slight bend to your elbows--they should absorb the shock. Shoulders down. Hands light.

Deerblue--same thing. Flat bars might be fine, but they might not solve the problem if the problem is with bike fit, core strength, and whether you lock your elbows or if you relax. Drop bars offer more hand positions than flat bars, and flat bars force your wrists into an unnatural angle.