How much are you kicking?
Try holding a flutterboard/kickboard/buoy between your legs and continuing the arm motion of front crawl.
This does two things.
Removes the leg muscles from the equation Keeps your legs up, which makes swimming MUCH easier.
The main trick in efficient swimming is to learn to swim downhill.
You want to 'Press the Buoy' which is push your chest (full of air) down, and use it as leverage to get your legs higher.
You sort of eventually end up swimming downhill a bit.
Another trick is to tuck your chin into your chest, imagining you are pulling your neck and spine far enough to tug your legs up.
(Its just a trick to teach you to lean forward and press the buoy, but the image works).
If you try to feel yourself pulling your legs up by your chin it can be effective.
Basically these are approaches to being more efficient.
sports.stackexchange.com/users/23/geoffc
This might seem a stupid question, but are you breathing out when your head is underwater?
Most swimmers that I see struggling to catch their breath are holding their breath when their head is in the water, and then trying to both breathe out and in when they turn their head above water.
There's not enough time to do both in crawl and they inevitably struggle to catch their breath, as you've got to breathe out before you can breathe in.
sports.stackexchange.com/users/850/mopennock
A good bet is that this is probably caused by being nervous, and you might be hyperventilating.
Given your fitness level, there's probably not a physical reason for getting out of breath - and I bet your breath very quickly returns to normal when you grab the side of the pool.
The good news is that this is 100% fixable - and you've already taken a big step: recognizing that there is a problem.
Time in the water, and maybe some coaching will quickly bring up your experience level, and your confidence, which should smooth out those nerves.
sports.stackexchange.com/users/36/Mike Hedman
When I started swimming seriously, about 4 years ago, I struggled with exactly the same problem .
It took me a long time to break the 100m barrier for front crawl , doing bi-lateral breathing.
Uni-lateral breathing was a little better
but I still struggled with a sensation of panic and being out of breath.
Once I had broken the 100m limit, suddenly it became possible to go much farther, as if there is a transition between how I would breath normally on land and how I needed to breath whilst swimming.
I still feel a slight pinch during training warm-up at this point, as if my lungs 'change gear'.
To beat the limit
I : slowed my swimming to a very gentle pace so as to be as relaxed as possible repeated 50m swims at first, with a break between each to recover, then 100m swims with a break eventually
, I felt I could attempt to go past the 100m and try for 150m Once I was getting to 150m, I very quickly made it to 200, 300 etc.
One of the key things that helped me was to beat the barrier
was expelling 50% of each breath immediately after inhaling, ie.
as my face re-entered the water, I already started to breath out.
The remainder of the air I slowly breathed out, through nose and mouth simultaneously, during the time my face remained under water, such that after 3 strokes it was all gone and ready for the next inhale.
As others have indicated here, breathing out is very important; when swimming you should be either breathing in or out but not holding your breath.
My coach explained the panicy feeling you get as being the need to expel the C0-2 from your lungs following breathing in, which is not something you'd instinctively think to do.
sports.stackexchange.com/users/2248/mungflesh
I'm in a similar situation, albeit I only run half-marathons and bike up to 60km.
When I started swimming I had the exact same problem, that I couldn't breathe according to what felt a good swim effort.
In the end I relaxed my swim speed as well as allowed my self to breathe every time I pulled back with an arm.
I really focused on my breath and feeling good in the water.
Everything else was secondary to being able to breathe calmly.
I train in a group, and before I would have to take a break every 50m and wait till my breathing came from a frantic gasping to a relaxed in-/exhale as while running.
After the change I swam slower, but could actually manage 1h30 of training, where previously I would be at the end of my strength after 30min.
Now, a few months later I can keep up with my group, and I breathe every 2 arm pulls, sometimes every 3 to switch breathing sides.
sports.stackexchange.com/users/1193/Rafael Emshoff
I am now starting to learn how to swim and have found that I have trouble swimming with my head in the water. Although I breathe every 1 or 2 strokes (front crawl) I found that my heart rate rises pretty high and soon I found myself gasping for air.  I am not out of shape; I run marathons and do long MTB races (80km) so I thought I can control my breathing during effort, but this seems a different kind of challenge.
Why do I get out of breath so quickly when swimming?
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sports.stackexchange.com/questions/548
sports.stackexchange.com/users/28/Gabriel Solomon
Relaxing in the water seems to be the key to improving breathing when swimming. Techniques that may help include spending more time in the water, slowing the swimming speed until you get the breathing right and using flutterboards to help perfect the arm stroke and breathing technique. There are a number of techniques to increase your tolerance. You could try swimming with your hands only, practice downhill swimming, increase your pace gradually and experiment with swimming positions. Part of the problem might be psychological. Familiarizing yourself with the water, and staying calm and confident are critical to your improvement. Moreover, breathing in and out constantly and evenly is key to swimming. You should practice your breathing rhythm and avoid holding your breath.
There are a number of techniques to increase your tolerance. You could try swimming with your hands only, practice downhill swimming, increase your pace gradually and experiment with swimming positions.  Part of the problem might be psychological. Familiarizing yourself with the water, and staying calm and confident are critical to your improvement. Breathing in and out constantly and evenly is key to swimming. You should practice your breathing rhythm and avoid holding your breath.