The most simple way of overcoming your fear of breath holding (and wipeouts) is that you have to know that you are prepared for every outcome.
Most of the surfers I have met in my life never actively prepared for getting caught under water. Sure, big wave surfers do, but your average weekend warrior just fears the wipeouts, but never prepares for them.
One of the reasons for this that it is not obvious how to prepare for it.
The articles in this section will teach you how to prepare for breath holds.