I generally second what Ollie said - writing is a nonlinear process and it is atypical (IMO) to not need to go back to the literature while writing at times, particularly as you identify other points you'd like to make in the discussion or when you are thinking of other potential explanations/directions, depending on your results.
OP, I get the sense that you are just growing your knowledge base at this point. At that stage of training, it can feel overwhelming (and YES, it is a lot different than a term paper. I tell my students this all the time - sometimes I ask them to count the references in the papers they are citing and then come back and let me me if they still feel my minimum citation requirements are unreasonable). It does get easier to some extent as you gain mastery over the area of the literature that you are writing within, although we always need to keep reading, staying updated, and conceptualizing things. I suspect that your mentor is giving you this to help you develop a) some writing skills and b) a way to problem-solve and develop your own process.
A formula that I think can be helpful for early graduate students is to do things somewhat systematically earlier on, because you don't know what you don't know. So pick some search terms and actually do a comprehensive lit review (rather than cherry-picking the studies that are most accessible to you). Investigate all articles that pop up and rule out the irrelevant ones based on some criteria. That gives you a semi-objective sense that you are at least taking an appropriate slice of the literature and aren't leaving something out. Then read the crap out of those articles and then put together your arguments, etc. I liken this to the process that most people have to go through when they do their masters thesis, heping them to gain some expertise in an area and actually reason their way to hypotheses.
Too much? Ok, fine, find some recent review articles from reputable journals and then dig up the more prominent studies to limit your sample. But recognize you are following someone else's reasoning and work, and eventually will need to be able to absorb and conceptualize the literature yourself. This might help you get started, but you want to read everything for yourself rather than just take someone else's word for it.
Another point I'd make is that there is something to be said about at least drafting the paper in a linear way. I think some labs are just throwing data at students, which they analyze with a technique they learned in stats last week, and then the mentor says "write it up!" Ideally, you let the literature determine what questions are important, how to generate hypotheses, and ultimately which statistical tests to use. So while in practice someone further along in their training or career might be able to start anywhere that they prefer (e.g., method or results), I think there is utility for training in going from start to finish just in terms of developing your conceptual capacity. Research in its purest form is a linear process - even if our creativity and ways of generating ideas or taking feedback are not always linear, the method itself is typically one of building on past work, assessing the results, and then building some more.
So when you get to the point where you do feel you have some expertise in the area and have a great internal sense for how the literature exactly informs your hypotheses and how those drive your method and results, then i'd say do what works best for you, and there have been some great suggestions. But if you aren't feeling as confident about that, then I think it makes sense to start from the beginning and be sure that you establish those connections well. Otherwise, you may very well be writing a results section without understanding your hypotheses at all - which is technically possible but not advisable IMO.
Good luck. It gets easier (and more fun) as you develop mastery. You might be banging it out in one sitting one day - but I wouldn't view that as the goal. Ultiamtely writing ought to be a thinking process that helps you to work through the specifics of your study.
