The first thing I will say is that bilateral breathing (that is, breathing every three, five or seven strokes) is not a complete necessity. What I like to be able to develop is the ability to breathe on both sides so that you have the option to switch.
To start with, what is stopping you breathing to the ‘other’ side? I would hazard a guess that your one sidedness comes from over extending to one side, giving yourself as much time and opportunity to take air in.
First and foremost focus on swimming smoothly and controlled. I like to get people doing lengths (or as far as they can control safely) without breathing – focusing on good posture, rocking the hips to get body roll/rotation, and a consistent and constant stroke with no stops or pauses.
If you can do this, then the breathing can follow simply (with practise). The difference between swimming without breathing, and swimming with breathing is purely turning of the head (hopefully!). So when you go to reintroduce breathing having done a couple of spells of no breathing/swimming with a snorkel, you want to aim for the only change being that turn of the head. Don’t forget that hip movement – that will be what makes turning your head to either (or hopefully both) side much easier.