Practice! Check out Khan Academy to learn concepts. Run through practice problems until it becomes second nature. Wolfram Alpha is a great resource for finding solutions (it's only helpful if you know what you're calculating, though!).

Make sure not to focus on just drills and solving equations. Word problems are going to help you much more with application of the concepts, even if they tend to be a bit more difficult IMO.

If you just want to focus on programming/web development, the most math you'll probably need is a decent grasp of algebra. If you're set on going further into CS, you'll definitely want a strong grasp of advanced algebra and trigonometry going into university (they may offer courses to help you get caught up if needed).

If you want to be a computer scientist in academia, then yes, you'll need 'higher-level' mathematics skills. Definitely more than the average software engineer will utilize.

Mindset & Approach

This may not sound like a very useful response, but you need to learn to teach yourself. It will prevent you from acting like a victim of poor instruction. Which is the attitude you basically led with in your second sentence.

In this regard, the one characteristic which has most aided me is self-criticism.
I had to learn how to judge whether I understood a topic and hunt for holes in my understanding.
Focus on thoroughness and being honest with yourself. This mindset will harden you and change the way you approach math.

My undergraduate program (electrical engineering) required TONS of math. If I had taken one additional math course, I'd have been eligible for a mathematics minor.
Additionally, my major 'focus' was electromagnetics...which is very equation/solution heavy.
In short, I've lived what I'm preaching. Past Calc I, no one was holding my hand. Professors assigned chapters + problems and gave a 1hr lecture (often of dubious quality) which spanned several topics. The rest was up to me.

Practical

One advantage you have over the average student is your coding ability. Leverage it.
I recommend that, after solving by hand, you code every single math problem you attempt. Turn it into a function. Plot it. Get creative. You can bet your ass, that's what you'll be doing all day as a comp sci academic.
Use whatever programming language you wish, but I recommend a combo of Python and Juypter-notebooks like the data scientists use.

One of my most favorite courses was an advanced radar course. The professor gave us raw data files output from a radar array and we used our math + programming chops to create a 3D video of upper atmospheric plasma events (causes airplane turbulence). It wasn't comp sci, but there are definitely parallels in the work flow.

Last of all. Seek out mentors and enjoy the ride ;)