High school math does not look much like research math.
High school math is highly concerned with teaching 'algorithms' that compute answers, like long multiplication, long division, the quadratic formula, completing the square, synetic division, u-sub integration, ...
Most mathematicians don't work with calculating things. Rather, they're more interested in _generally_ characterizing how objects behave (and _proving_ that that characterization is actually correct)
For example, the definition of "continuous" you learn in pre-calc is probably something like: if you compute a function's limit, you get the same thing as computing the function.
The topologist's definition is that the function's inverse map preserves openness.
These are basically unrecognizable as being the same thing, but in fact they are. And the second doesn't actually reference calculation or even anything identifiable as numbers!
I believe this is a significant cause of not understanding "why" despite understanding "how". As an example, I remember in pre-calculus we spent at least an entire unit on partial-fraction-decomposition. Partial fraction decomposition is a strange algebraic manipulation that is basically only important to integrating rational functions -- something you don't do until at least a year later, and which doing by hand is basically pointless because computers are better at it than humans.
They also require very different kinds of intuition. Building intuition is hard, but important for people studying math. Working through many different examples and problems and applications can help you build intuition.
But I don't really have any advice for how to make math stick better, but just want to share that the mathematics you learn in high school is not representative of the entire field.
If you ever want to get a taste of "higher math" without having lots of prereqs, point-set topology and group theory are two (very different!) fields that are also fairly accessible.