Just read up a little, There are two types of similar conditions caused by nerve compression in the wrist area. Carpal Tunnel Syndrome, and Ulnar Nerve entrapment (Cyclist's Palsy, or Handlebar Palsy). It seems I may have the Carpal Tunnel Syndrome. My wrists where that "tunnel" is hurts for a long time now, could be a few years, but I just happened to read about these two types of injuries, and it's more like Carpal Tunnel Syndrome, according to How do carpal tunnel syndrome and ulnar nerve entrapment differ? It's almost certain from the pressure of holding the flat handlebar for extended time period. It also gets worsened from swimming.

Are these conditions common among cyclists? Should drop handlebars be better than flat bars?

Here is a nice video I found: