Michael Phelps was a world record setting butterfly swimmer and he would breathe every stroke. He breathed shallowly but frequently (more instructions in the link or just watch him on YouTube or similar).
Another option would be to simply skip a breath every time you feel you don't need one. I.e. rather than breathe every stroke or every two strokes consistently, breathe intermittently. If you feel short of breath, then breathe. If you don't, skip that stroke (and definitely breathe on the next stroke). This might end up with breathing two strokes and then skipping one. But what I'm saying is that rather than breathe every stroke whether you need it or not, breathe when you feel you need it. If you feel out of breath at the end of your forty-four strokes, then breathe more next time. If you feel like you have too much breath, breathe less often.
Phelps' insight was that breathing shallowly every stroke gave him plenty of breath without slowing him down too much. I would encourage you to try it. Lift your head less and take shorter breaths. If that doesn't work for you, try being more intermittent. Don't breathe every stroke but don't skip every other stroke either. There's no score for consistent breathing in swimming. Work out something that feels right to you, not that matches some arbitrary metric.