best rotation helps enormously with the head turning not lifting
When you're breathing out, have the face down and imagine the tip of your head has a laser on top and you have to have that lined up with the end of the pool.
Relax your neck entirely so that the water holds your head.
You can practice doing a few push and glides to see how relaxed head/neck feels. It helps a lot.
On the crawl stroke Keep the hands in line with your shoulders when entering the water, extend the hand and arm and reach and open the shoulder muscle to make that reach stretch. That automatically helps you rotate.
Keep your head still, relaxed and down in the swim stroke, then when the breathe stroke comes your head is allowed to turn with your shoulder and that puts your mouth pretty much in a bow wave space to breathe in.
The other way to work on breathing is to take a float with one arm keep your ear on the arm and just turn the head to the side, eye line no higher than the lane rope. It's dull, but it commits it to muscle memory and that's what you're aiming for.