Yeah, I think I was struggling with this aspect. I recall my professor saying something like "it's not that general relativity is correct and newtonian mechanics was wrong, it's that when a general relativist says 'mass' they are talking about something different than when a Newtonianist says 'mass' - after all, you have to measure mass differently in those two things, they behave differently, etc. It's more that general relativity doesn't say anything at all about Newtonian mass." But I never bought that (even if they were talking about different things, it seems like they were trying to talk about the same thing), so I figure I'm missing something from Kuhn's argument.
If you believed the world is flat you can still get from one village to the next one and you wouldn't fall of the earth. It's true enough for what it is trying to accomplish. If you want to navigate longer and longer distances or go to the moon however this believe will meet it's limits.
The primary thing people struggle with in general with science and philosophy of science is actually more fundamental in other parts of life to which is Truth.
Popper thought science helped us approach the some objective Truth. Kuhn realized (and I agree) that truth is always depending on the context in which it's defined.
Personally, I learned to think of scientific ideas along two axis - precision and usefulness. To reuse your professor's example, in so far as you accept that Einstein and Newton were really talking about the same concept of "mass", general relativity is much more precise than newtonian mechanics - but newtonian mechanics is much more useful than relativity, is the sense that it's precise enough for 99.9% of applications (including sending spaceships around the Solar System) while also being significantly easier to use.
