My dad still doesn't know how to swim, at age 76. His brothers tried to "teach" him, by throwing him into the irrigation ditch by their home. A neighbor found him unconscious, floating through the fields.
My dad has big, knotted hands. He's not afraid of spiders, or of heights, or of death. Heck, he's sailed across oceans. He is a strong and stoic Russian, but he's still afraid to swim. He lives in Hawaii and he won't go into the ocean.
My mom taught us kids to swim before we could walk. Swimming, for me, is like being free. I love to swim and there is no other activity that gives me the same feeling. The quiet calm of water, the feeling of being held by the ocean; It's beautiful.
But, I bet I'm afraid to do something that doesn't scare you a bit. In fact, I'll tell you a secret. I'm afraid of the dark - or rather - what could be in the dark. Imagine that, a grown woman with two small children, afraid of the dark!
Courage, is doing something even though we are afraid. For me, that might be investigating a sound in the dark (baseball bat in tow?). For you, that might be putting your head under water. We all have the ability to be brave, but it's hard and it takes practice. Take it slow, try to have fun, go to the pool with people who don't make fun of you and be gentle on yourself. We're all afraid of something. But, we get to choose to accept that fear and live in it, or to work through it and rise above.
I suggest teaching yourself (being taught) in the same way you would a child who felt uncomfortable. Slowly, with a lot of laughter and play, get your head under water and talk yourself through it. Reward yourself for small successes. Once your feeling more comfortable, get help with your technique. I suspect the biggest hurdle will be finding confidence and the rest will follow more easily.