To understand the rotating disk in special relativity, I believe you need some background material that is not high school level. The place I would suggest to start getting the necessary additional background would be to read about Born rigidity. I would suggest modest expectations - realisticaly, you are more likely to learn enough to know how difficult the problem is, rather than learning enough to solve it.

Back to Born rigidity.

Born proposed a simple notion of rigid motion in special relativity. In lay language, Born rigid motion is any motion that leaves the distance between all pairs of points on a body constant as it moves.

A wiki reference on the topic: https://en.wikipedia.org/wiki/Born_rigidity

Erhenfest, using Born's original notion of rigidity, showed that it was not possible in the frameork of special relativity to make a disk start or stop rotating without violating Born's original criterion for rigid motion.

That's not the end of the story, but to go further will likely require much more than a high school background. One general possibilities include - using a more relaxed notion of what it means for an object to be rigid, notions of which are mentioned in the wiki article I cite.

wiki said:

Several weaker substitutes have also been proposed as rigidity conditions, such as by Noether (1909) or Born (1910) himself.

A modern alternative was given by Epp, Mann & McGrath.

I'm not personally familiar with any of the "relaxed" notions of rigidity, I simply know they exist.

Other approaches might involve keeping Born's original notion of rigidity, but specifying a material model as to how things stretch. Greg Egan took this approach in a non-peer reviewed paper, which has apparently changed since I last looked at it, at https://www.gregegan.net/SCIENCE/Rings/Rings.html. It's also not high school level, I'd put it at the graduate level.

Objects in classical Newtonian physics deform all the time, and their behavior can be analyzed. But even without relativity, this would likely be college level rather than high school level. Mechanical engineers deal with this all the time. I'm not aware of what texts they use, and their approach is different enough where you may not get enough leverage out of it for it to help you understand the relativistic formulation.

To summarize. Simple notions of mechanics based on idealized rigid bodies simply cannot handle the relativistic rotating disk. I believe you will need some notion of how to handle objects that can deform to handle the problem. This may leave you feeling sad if your entire understanding of physics is based on rigid bodies which are unable to deform. The solution is easy to state but difficult to cary out - learn enough physics to handle bodies that are not necessarily rigid, bodies that are able to deform. This will probably be beyond the high school level, though. The typical approach to the continuum problem involves partial differential equations, where the "rigid body" approach gives ordinary differential equations.

A very simple example of the continuum problem, one that might be approachable at a high school level, would be to understand how, when you push on one end of a bar, the bar does not instantaneously move in a rigid manner, but how a wave propages through the bar at a characteristic speed of the material, a lot like a wave travelling along a slinky. The slinky analogy is particuarly fun and useful because the speed of propagation of a wave through even the most rigid materials (often called the speed of sound in the material) is MUCH less than the speed of light. I'm not sure even the "wave equation" is truly at high school level, though, because it requires some understanding of partial differential equations :(.