Sit-ups during the PFT (Personal Fitness Test) are conducted with a partner holding the ankles/feet of the person carrying out the repetitions. These not only work the abdominal muscles, they also work the hip flexors.
A good training tip is to perform 50 (the minimum you need to pass said PFT) at the end of every workout, assuming you do train a few times a week. It doesn't matter how fast you do them, just that you make sure that each one is a good repetition. Your body will get used to doing 50 repetitions, and then you can work on decreasing the time it takes you to do those 50 repetitions. Not the best way of training, but if you're just looking to improve your sit-ups, it'll work.
This is not normal, you need to seek medical advice. If you are a soldier that does not like to go sick the minimum you should do is buy a good wrist support (you get what you pay for). if your wrist hurts whilst wearing the support then you need to report this. if not carry on for a couple of weeks then try with out the support. if it hurts then, then again you need to report it.