Stretching exercises and more swimming will give better flexibility to allow a more horizontal body position. Push your chest down into the water and feel like you are swimming downhill. You may not have a rhythmical breathing technique. Blow out air into the water when your face is in, so when you turn your head to breathe your lungs are empty. Do this while standing in waist deep water.
Rhythmical breathing is why swimming is good for asthmatics.
Then try swimming slowly. Your head might be too high too. A correct head position is when one ear is in the water when you turn to breathe. There are other reasons for being out of breath - maybe you are not fit, or you are tense. Do a little bit of swimming have a rest, then do more.