Is there a simple layman way that I can use to explain the incompatibilities between quantum mechanics and (general) relativity to high school students (people with not much knowledge of the intricate math of quantum mechanics and (general) relativity)?
For what it's worth, there's a simple argument which explains the need for quantum gravity, using just dimensional analysis:
Equating the two, we can derive a special scale called the Planck scale. An imaginary particle with Planck mass has a Compton wavelength and Schwarschild radius of about the same size, so for such particles (i.e. when we deal with such energy scales) both general relativistic effects and quantum effects become strong -- this is why we really need a theory incorporating both.
As for why combining the two is hard:
Probably no simple explanation. It is however important to emphasize that the incompatibility applies only to general relativity. The special relativity and quantum mechanics are very compatible and were luckily married many decades ago, giving birth to the quantum field theory which is an incredibly successful framework in which physicists built the quantum electrodynamics, quantum flavordynamics, quantum chromodynamics and the whole standard model. The whole modern quantum physics would not be thinkable without combining quantum mechanics with special relativity.
General relativity is a different case however. The root cause of the issue is rather technical, so laymen terms do not reasonably work here. Basically, when you try to quantize gravity, you get nonsensical (infinite) results that cannot be remedied. A solution to this problem is not yet known.