I am self-studying physics, and I just finished Linear Algebra by Howard Anton, Vector Calculus by Susan Jane Colley, and Differential Equations by Boyce DiPrima.
In order the learn the mathematical formulation of quantum mechanics and general relativity, I need to take a second course in linear algebra and learn real analysis, then topology and metric spaces, which will be followed by differential geometry. This is the route mathematicians take when studying differential geometry and advanced linear algebra.
But I don't want to study proofs. I want to learn the mathematics behind physics in a rigorous way and understand all the concepts thoroughly, which is why I avoid any "mathematical methods" book, but I just simply can't stand proofs, especially the trivial ones. For example, a lot of the content in Sheldon's Axler's linear algebra is relevant to quantum mechanics, but the entire book is proof-based and that's not really helpful. Similarly, proofs are at the heart of Real analysis but ain't gonna help me in physics. I am really confused because I can never study mathematics from the "mathematical methods" books because they just take away all the beauty of mathematics, but don't want to study proof-based books either. I am looking for books that are mathematically rigorous, and explain all the concepts thoroughly, but aren't proof-based.
Personally, as others have stated, I disagree with this idea that you can learn mathematics without understanding proofs. Every book that is mathematically rigorous, and every treatment of some field which is mathematically rigorous will rely on proofs in some form or another, as that is what makes something mathematically rigorous, the act of proving things.
That being said, if you wish to learn physics from a mathematical perspective, there are a number of good books you may enjoy:
It is important to note that these books are not devoid of proofs, and that the last two books have a heavy mathematical flavor to them, but I believe they are all aimed at both mathematicians and physicists (maybe less so the third one) and thus may help bridge this gap you are experiencing. In particular, since the books are focused on physics, the proofs may feel less like proofs, and the text surrounding the proofs has a great deal of exposition.
At the end of the day though, if you're not good at understanding physics without a rigorous foundation in the math the physics is using, you really ought to become comfortable with proofs.