I have no idea because Im not a swimmer, but Mrs FD says correct technique is to only breath to one side, not alternate sides.
she’s wrong – pretty much all good swimmers breathe on both sides. Breathing on just one side is a recipe for an unbalanced stroke, which leaves you wasting effort correcting it if you want to go straight.
There are times when you want to breathe one side only, like if you’re sea swimming, you tend to breathe down wind, and if you’re in rapids you might want to do it only one way depending on how the currents are, but for normal swimming, both sides.
The original question, lots of people have covered it already, you say what you do wrong in the question itself – you should never be holding your breath, breathe out under water, then only breathe in when you turn your head to breathe. At swim coaching sessions I’m doing at the moment they tell us to think ‘bubble, bubble, breathe’, on each stroke, where you are breathing out for each ‘bubble’, and in on the ‘breathe’.