How to learn without looking at solutions? (real analysis)

You shouldn't feel guilty for looking at solutions. Your struggle is something that lots of people, including myself, share. The truth is that no one is able to solve all problems they are ever given on their own, because often, the problems might involve a new idea you've never seen before, or a technique you wouldn't have thought of, and these are things that you have to pick up along the way and cannot be expected to invent on your own.

My advice is this. When you're trying a problem, write everything down. Don't be afraid to waste paper scribbling down notes which don't help towards solving it. Very often, the solution to a problem might be easy once you've written your progress down, but the problem might seem completely opaque if you haven't even dared to try anything. So, keep trying, and keep writing down whatever little progress you have made. This should give you a sense of what exactly is hard about the problem, and what exactly is the hurdle you can't get over.

After you've been trying the problem for a certain time $P$, if you're still getting nowhere, it is fine to look at the solution. But after doing so, close the textbook or browser and just try to work through the whole proof yourself, without reference. This process of writing everything down will make the logic "flow" and seem natural to you, and ensure that you really understand what the proof is. It is imperative that you've tried the problem before this, because then, you gain a lot through finding out how exactly the thing that was stopping you from solving the problem is overcome in the actual solution. This is how you learn and gain new insights to improve your skills.

For me, I've found that this time $P$ which works best is around $30$ minutes for routine homework-style problems, but this time might vary for you. Try this out and find the best time that works for yourself.

I have the possibly unpopular opinion that it is fine to look at solutions, as long as you attempt the problem first. There is an art to balancing this, but if you are the type of person who feels guilt about it, you can probably trust yourself to distinguish between cheating and just wasting your time. Every field has its own proof techniques, and many professors forget about the learning curve when they teach new material. You will probably find that, over time, you will have a much better intuition for how to either approach a problem, or to know that it is out of reach.

Also, once you reach proof-based math, you are forced to fully understand and recreate the proof on your own regardless. If you fail to figure a problem out, going through someone else's solution is still instructive. It will never be as good as coming up with it your self, but theres two sides to every coin. Or one if you're a topologist.

I also agree that objectively is not bad to look up solutions per se. I already agree a lot with the answers given so far. Maybe let me add this point.

Studying mathematics is no a sprint but a marathon. So what happens on the first 20 km doesn't really say much about what happens after 30 oder 42km, as long as you stay true to yourself and keep trying. I study mathematics for a couple of years now and every single new semester I am amazed of the new mathematics that I am exposed to and in the beginning it is always quite hard and I feel lost. It gets a bit easier because I went through this experience a lot that you described where one is simply shocked what a certain solution to a problem looks like. And even though I think I am not that talented in mathematics through a lot of effort I became quite decent at it. And maybe to stay in the metaphor: "This kilometer is quite nasty but hey, so was the one 10 kilometers ago and then it became a bit better again. So this might past as well".
So here I wanted to tell you, pull trough! Don't give up. You learn this stuff for you, don't get jealous or frustrated because this or that person seems to have it much easier. Everyone can contribute and again for that it really doesn't matter when or if you struggled, because in this everyone does.

Let me also stress this point about the learning curve. I have some experiences with other fields and I can say that mathematicians are a super nice and relaxed and a smart bunch. But when it comes to the pedagogy of how to teach this field I feel many instructors lack a bit of inspiration!

Finally, here is something a professor of mine told us that I found super helpful:

"There are (at least) three types of students: Some very few students look at a homework sheet and solve it within a few hours. Those people exist. Then there are students who work on the homework the whole week and struggle but in the end they get most of it. And then there are those who struggle for a week and don't quite get there on there own, but when they see the solution they understand. I think all of these types can be very successful students."