ETA: I was composing this before I saw your comment about the textbooks, RRiley99. It's rough when the prof doesn't follow a textbook closely, but even so, you may find that the stuff in the book can still provide a useful counterpart to the course material. It may have an alternate perspective, or it may have essentially the same material, just in a different order. So it may be worth reading the text anyway, even if it doesn't perfectly correspond to the course. You may also wish to ask the prof for guidance on what parts of the text you can refer to. And even if the text is completely hopeless, many of the techniques I describe below can apply to the course notes as well.
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The "no textbook" thing is surprising to me, too, since there are plenty of decent textbooks out there on those subjects. If you don't have a textbook (or a good textbook) for any particular class, you may want to acquire one. I don't know exactly what level your courses are being taught at, but the ones I used in those areas were Jackson for E&M, Goldstein for classical mechanics, Liboff for intro quantum mechanics (although I think Griffiths is more commonly used), and Merzbacher for more advanced quantum (although I think my immediate predecessors used Cohen-Tannoudji). I'm not going to say those are the exact optimal textbooks, but they weren't bad, and they'll definitely be better for you than nothing. Your school library will probably have most of these books or decent equivalents.
There's a lot of stuff in a textbook that you can't find easily on the internets, and that lectures or official course notes may skim over. I would recommend trying to find a book or set of books that you can read along with each course as best as possible. (Check the indexes against the course syllabus or get advice from the TA or prof if you can't figure it out yourself.) The worked examples in these books are particularly good resources --- when you come to one, sit down with your pencil and paper and work through every step yourself, trying to write up a full explanation of the logic as if you were going to hand your work to someone who hadn't read the book, and use it to teach them how to solve the problem.
But don't ignore the rest of the book in favor of the examples. Understanding the derivations of the various equations you're using, and the physical reasoning and assumptions underlying those derivations is essential to being able to use them effectively. (True story: I used this trick to break the curve on a closed-book exam one time, because I could rederive some of the key equations I needed, and others couldn't.) So make sure you follow along with the derivations and explanations too, and keep careful track of what's being assumed at each point (and also of the overall motivation for the derivation). Following the derivations will also help expand your library of mathematical tricks, and make it easier for you to recognize situations in homework problems that call for a particular trick. These techniques can also help you in studying your lecture notes and the official course notes.
The second thing I'm wondering about is that I don't hear any mention of working with other students. Do you have any option of discussing the problems with your classmates as you're solving them? Those kinds of discussions are often much better for solidifying your understanding than just asking a TA questions, because they give you an opportunity to formulate an approach and argue for it against your friends, in a situation where none of you is really authoritative. That means that you have to actually make a good case, and that it's important to try to pick holes in your and your friends' ideas, because any of you could be right or wrong. The more practice you get constructing and destroying solutions in this kind of environment, the better you'll be able to do it when you're on your own.
The third thing I can see that might be causing you problems is that for classes at the level you're describing, 15-20 minutes is nowhere near enough time to be spending per problem, even (especially) if you're totally stuck. I don't know how many problems you have on each homework assignment, but I remember having assignments of around 4-5 problems that would take me a day of steady work for the whole assignment *if* I didn't get particularly stuck on anything. Getting stuck meant things took *much* longer. I know you probably have a lot of work to do, and not a lot of time to do it in, but I'd strongly recommend that you figure out how to allocate more time for flailing hopelessly at these things, because you'll never learn how to get to the "Oh riiight" on your own if you always let the TAs or prof do that step for you. And once you *do* start getting to it on your own, it will stick in your head far better than something somebody else just told you. I am *not* saying that you should never ask for help, or that you should annoy your TAs by putting off asking for help until the night before the assignment is due, but I think you will find that you'll be better off in the long run if you consider asking for help on something as big as, "How do I start?", to be a very desperate resort, and do everything in your power to avoid it.
Fourth, what Vagn said @4: if you *do* have to get major "how do I start" type help from an instructor, you've got to do extra work to make it stick, in a way that you may not have to do with something you figured out on your own. In addition to trying to replicate your solution at later intervals, you may also want to look for similar problems in textbooks or online and try to solve them too. Looking for alternate solution methods is sometimes good too, as well as trying to see what happens if you change some of the conditions of the initial problem. (This can also help you check your work/understanding in general --- does changing a particular parameter or assumption change the answer in the way you expect, and if not, why not?) The more variability you inject into your problem-solving environment, the more robust your eventual understanding will be.
Anyway, there's no one perfect solution to this issue. Part of the problem here is that you've hit one of the transition points in physics education where you suddenly have to do a *lot* more work to keep progressing, and it's really hard to figure out how to adjust, or even to recognize that you *have* to adjust. And of course, nobody really tells you this, so it's very easy to find yourself struggling with no idea why, thinking you must be the only one. But the bottom line is, this is the point where everything starts to take a heck of a lot longer than it used to. You're struggling because it's actually objectively hard and time-consuming. So be patient with yourself and the material, allocate as much more time to your studying and assignments as you possibly can, and you'll get there eventually.
