There are a couple of things at play here. As the previous poster mentioned, you should try to slow down your pace so that you aren't so out of breath after one lap. Even if you think you are going slower than anyone possibly could, slow down some more. It's okay. Get the running motion into place for your body, ease into it.
Second thing your could try, is to start running intervals. Run for 60 seconds, walk for 60 seconds, til you have run the whole mile. Then (gradually over weeks), increase your running and decrease your walking until you are running the whole distance.
I read once that when we start to run, either our legs or our lungs are stronger. It does take time for the other to catch up, so it sounds like your legs are wanting to go farther, but your lungs aren't quite ready yet. Slow down to give the lungs a fighting chance!
You can do this!
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