Hi, I'm new to swimming but have managed to get the breathing working correctly.
It certainly took a while but some key things that got breathing to work for me were:
1. Hang the head and always look down at the bottom
2. When rotating from side to side, allow the recovery arm shoulder come up over the surface, then the arm and then just before hand re-enters the water rotate the head back down with the body i.e. try to keep the head body aligned at all times.
3. When the head was rotated with the body feel like your head is resting on the water (kept aligned) and "see" the side of the pool your face is facing.
Needless to say I had to ensure my fill of swallowing water as my timing was poor at the start but eventually it came right. I use to try just rotating the head to the side to ensure I could see the side of the pool I was facing and NOT take any breath... and this with mouth open.
Whatever above slowing your strokes down you certainly have longer to take a breath than is thought! I also tried counting - as my head was breaking the surface 1 and 2, when I heard myself say 2 then I rotated back down into the water.
Doing this has eventually got me feeling very comfortable now.
Mind you, at the very beginning I had an AHA moment when I realised in 4 foot of water I couldn't drown (unnecessary panic) as all I had to do was stand up!