As a long-time swimmer, I think this is just a coordination issue - a bit like trying to rub your stomach while patting your head. When you think about one part, the other tends to stop until you 'train' your brain to do both.
Rather than using a pull-buoy as suggested by the other answer, I would suggest a different exercise: use a kick board held out in front of you with one arm, and practice using the other arm to pull and breathe to that side (pick the side you find most comfortable breathing on). Kick with your legs the entire time.
Using a kick board in this way has several benefits:
- Your body will stay flat on the water which makes kicking easier.
- Doing single arm pull is less to think about so you can focus on the pull-breath-kick mechanic.
- Additional buoyancy towards your head will make breathing easier.
Once you get used to doing it on your dominant side, switch arms and do the other side (it's important to breathe both sides if you want a balanced stroke), then try doing full stroke again and see if it has helped.