Yes! It is normal. Have you been a swimmer before? Or are you entirely new to swimming? People who can't swim and swim and swim are not necessarily out of shape. They're more likely out of practice and inefficient. The best thing you can do is spend lots of time in the water! Work on your breathing rhythm (this is usually a big hurdle for new swimmers). Something that helped my husband, who recently took up swimming, was pretending his chin was connected to his shoulder. This way you're not turning your head too late and shorting yourself on oxygen. Make sure you're also exhaling IN the water and not waiting until your head is out for a breath. Experiment to see what works best for you as far as how often to breathe. You might need every other stroke, you might only need every 3rd or 4th stroke.
Have you learned to do a good streamline position? When you get in the pool, are you just doing freestyle? Or are you using a kick board or pull buoy? What do your workouts look like right now?