I’m finding that my left hand gets fatigue and discomfort across the back of my hand. Is this common and what are the possible suggested solutions?
Are you gripping the flute tight?
This can happen due to wrist extension (where the back of your hand is at a fairly high angle to your wrist (often almost 90 degrees in some flute players).
I’ve learned to try to keep a more neutral position by adjusting my arm and wrist so the wrist is in a neutral position and the back of my hand is roughly in line with the rest of my arm. It may feel very unnatural at first, but I’ve been playing this way for years and rarely get the kind of fatigue or pain you’re describing. My grip is similar to what’s commonly called the Rockstro style, but my left hand fingers are a bit more relaxed; I’m hitting the holes more with the pads of my fingers than the tips, but not as much so as if you used a piper’s grip.
Brad’s suggestion is important.
It is certainly not healthy to have a cocked wrist while also pressing and gripping.
I experience a similar issue. This eased with time either because I learned to relax or my muscles became accustomed to the position. Probably more of the former than that latter.
If you grip the flute with your left hand (that is, if you squeeze the flute between your left thumb and left index finger), your left hand will fatigue in short order.
Instead, try supporting the flute with three points of contact - your chin pushing out, the first knuckle of your left index finger pushing in and the tip of your right pinky pushing out. You are clamping the flute between your chin, knuckle and pinky tip. Now put your right thumb directly below the flute to counter the force of your fingers pushing down on the holes. You are not gripping with your right thumb - it’s acting like a post to support the flute. That’s the grip I use and the one that I see most other players using.
Some people like to push out with the right thumb, which is called the ‘Rockstro grip,’ after the author of a 19th century flute tutor. I’ve never mastered it and don’t feel the need.
I use the grip just described. About the left thumb. Many people
grip the flute too tightly with their left hand. That may well be
causing your fatigue. The remedy is to practice with your left
thumb dangling off the flute. You don’t need it to do any work,
or very little, in holding the flute if you use the grip just described.
If/when you find yourself gripping the flute too tight while practicing,
dangle the thumb.
I’m a beginner, & found I was gripping the flute too tightly, which was making it difficult to finger the notes & get good sound.
After forcing myself to relax my grip, I’m doing better now. 
My hand-wrist angle was quite steep. I’ve experimented with lessening the angle to a more neutral position as you describe and this seems to help. I use the standard three support points; chin, inside of my left index finger bottom joint, and pinkie, and don’t ‘grip’ with the left thumb and index.
Where is the ideal position for the left thumb? I still vary between under the flute, either higher relative to the index or between index and middle fingers, or around the side of the flute. I feel tension discomfort in the back of my left hand after about 5 minutes playing.
A good place for the left thumb is just barely touching the Bb key, if you have one.
If you don’t have one, put the thumb where you would if you did.
My thought on this, bearing in mind I’m a beginner, is just let it sit where it naturally falls, that is, don’t force it into any other position than it naturally sits.
That’s where I put it. This way, if you do have a Bb key, all you need to do is roll in your thumb slightly to hit the touch of that key.
I can update that I’ve been looking at how to remove this l/h discomfort with my flute teacher.
The solution has been a small but significant adjustment of l/h position so that the flute contact point is very slightly lower on the inside of the left index finger. Also adjusting so that the contact point for my left thumb is toward the very base of the pad making the thumb point outward toward my body.
These tiny adjustments seems to have resolved the discomfort.
The solution has been a small but significant adjustment of l/h position so that the flute contact point is very slightly lower on the inside of the left index finger. Also adjusting so that the contact point for my left thumb is toward the very base of the pad making the thumb point outward toward my body.
Umm, yes. These adjustments can be discovered sooner with a one-on-one teacher!
You’ll also notice that your left index finger will now have a lot more speed for playing cuts, because the knuckle at the hand is much more agile than the 2nd knuckle. (These joint probably have specific names, but I’m not a doctor. Not a music teacher, either.).