I'm not sure if "overturning" general relativity is the appropriate description. We may well find a broader theory which contains general relativity as a limiting case, just as general relativity has special relativity and Newtonian mechanics as limiting cases. With the plethora of experimental verifications of general relativity, however, I wouldn't expect to see it completely discarded in the way that, e.g., phlogiston theory was.

Oh, I'm not calling mathematics more or less reliable than science. I'm saying that the ways in which one would overturn an established useful theorem would be very different from the ways in which one would overturn an established scientific theory. Another way in which mathematics is more reliable is that bias is irrelevant. Scientists have to disclose their conflicts of interest because it's easy for those conflicts to interfere with their objectivity during data collection or analysis, and so others must pay special attention. Mathematicians don't need to because all their work can be contained in one location, and can be checked in a much more rigorous fashion.