I can't breathe bilaterally for very long -- maybe one length! I'm still a newbie swimmer though.
My hope is that as my balance gets better, drag decreases, movements get more relaxed and efficient, that I will eventually move to bilateral.
But for now, I have to breathe every other stroke. I'm not trying to break any speed records, but just swim slowly for as long as I can. My record so far is nearly an hour swim non-stop. I swim breathing to one side on one lap, then breathe on my other side on the returning lap. This way at least I don't get lazy breathing on my 'easy' side all the time.
You might give breathing every other stroke a try for the very same reasons.