I am, at best, a novice mathematician. I started teaching myself the subject while writing my thesis in computer science. I find that I have a strong urge to prove every relationship or formula that I come across while studying. I routinely find it hard to accept or understand a relationship that I cannot prove and often go to great lengths to do so. Is this practice normal or recommended or otherwise?
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It is very highly recommended. Although when you begin doing things it is sometimes more fruitful to simply gaze at some ideas and accept them to let blossom more amazing ideas, at some point you just gotta start thinking and that is where foundation becomes important. It will never hurt you to start making some rigor go into your mind, as long as it does not stop it from finding new ideas. I believe that it takes much, much work to have both rigor and intuition, but that is the price to pay, because it's the best thing to do. Simply focusing on rigor won't get you anywhere though ; proving things is one thing, understanding why they're true is another.
Hope that helps,
If you just mean that you want to look up, read, and understand proofs of every result you use and that you can't prove for yourself with a bit of effort, then this is fine and admirable. In practice you might still want to take some shortcuts. For example, most people would forgive you for not hand-checking a proof of the Four-Color Theorem. The trick is to find a good balance of how long you try at something for yourself before "giving up" and seeking an existing proof. A tutor or textbook author can guess for you, by already knowing the proof, whether it is likely to be within your reach, or whether certain specific hints can easily send you in the right direction. So if something is left as an exercise in a textbook then by all means attempt it, because the author thinks you can do it.
If you mean, discover your own proof of every result you use and look up nothing, then this is the equivalent (in computer science terms) of refusing to use any compiler, OS, or web browser that you didn't write for yourself, alone, avoiding looking at the source of someone else's. It will severely hamper your progress. In practice you will want to use results whose proofs were significant pieces of work by people probably smarter than you and certainly more specialized in their field of study. You will find that there are results you want to use for which you fail to find a proof at all on your own. You cannot reasonably hope to reproduce over 2500 years of work since Pythagoras ;-)