What is the experimental evidence that suggests space-time is curved? Sorry, if this is a stupid question. tl;dr: Skip to the last paragraph for the question.
I know curved space-time is one way to express gravity and so far it seems to work well. 
I still need a lot of work to fully understand the Einstein Field Equations, and how do they arise and how to derive them.
We make theories to explain observed phenomena, so I looked at some experimentally confirmed consequences of general relativity, like the gravitational lensing, gravitational red shift and orbital precession, etc. But if I think about it again, these doesn't immediately mean the space-time needs to be curved. 
Some of my (maybe stupid) thoughts:


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*Gravitaional lensing: photons are massless particles with finite momentum, if you apply force them, their momentum will change, so their direction of movement. 

*Gravitational red shift: photons lose energy as they climb out of the potential well, and De Broglie relations tell that less energy means less frequency, so redder light. 

*Gravitational waves can be derived from the GEM equations which are just a copy of the Maxwell equations with a sign flipped so the gravitational charges of the same sign attract. Although they arise as a weak field approximation of EFE, they look very reasonable.

*Orbital precession says Newton isn't right, gravitation isn't exactly an inverse square law, it needs to be patched to describe the precession. Just like other things like kinetic energy and so on needed to be patched with special relativity.

*Black holes: the most important observation about them is that they are black, stars at the center of the Milky way orbit a void with crazy speeds. But this doesn't sort out the possibility that it's just a compact object with extreme gravitational red shift.

*More stupid ideas would follow... 


In quantum mechanics, we have the Bell theorem which rigorously proves that local hidden variable (cellular automata like) theories cannot explain all phenomena in QM, so making them no-go theories. And Bell theorem have strong experimental evidence from various CHSH experiments. So I don't even try to bother to think about local theories when thinking about QM, because the entire class is busted by the Bell's theorem.
Now the question: 
Is there a similar mathematical theorem with strong experimental support that proves that no flat space-time theories can explain all consequences of general relativity? So making flat space-time theories a no go, so I will be convinced not to try to bother coming up with stupid ideas like those above...
 A: Feynman had an example of a disk with a temperature distribution and say some ants that live in the disk where it is hotter nearer the edge. If their rulers changed sizes with the different temperatures at different locations then they might lay some identically manufactuted measuring sticks down and notice a circle about the center of the disk requires different than $\pi$ times as many to go around the circumference than to go through a diameter. This is because the disk is hotter near the edge so the measuring sticks out there expand and so fewer are needed. These ants might think they live on a curved disk, but we know they don't, we know that their measurement devices just aren't measuring actual distances. And if they moved their devices fast enough to not have time to get to equilibrium with the disk they would notice that as well as other effects.
Could the same thing happen to us? Yes. Our meter sticks might not measure proper distance and our clocks might not measure proper duration. However when we postulate that they do and then make a theory about what these distances and durations are then we can use a small number of physical insights to single out a theory with a small number of parameters that agrees astoundingly well with what we actually see.
So the world acts as if it is curved, and if we make a theory about the actual measurements we make can make a curved theory fit the data and that allows us to interpret the measurements to be of actual distances and durations. Could we be wrong like the ants? Sure, any science could be wrong if repeated measurements do not hold up or if new measurements in new domains or to new levels of accuracy don't hold up.  But then we still have a nice theory that a new theory has to reduce to in the appropriate limit where the old theory was good.
Any manifold with a straightforward topology could be modelled as a section of $\mathbb R^4$ with possibly some parts removed. And maybe there are flat space distances in that section of $\mathbb R^4$ and the things we measure with our clocks and meter sticks are complicated functions of the real distances and durations that only looked like results from a curved manifold in some limit.
But this could happen with Bell's theorem too. Maybe some kind of ER=EPR holds and so entangled particles are never truly spatially separated so it just isn't possible experimentally to have entangled particles that can be manipulated independently. So if we were wrong about which particles are far enough apart to test locality, we don't know about locality. Or if there is a kind of superdeterminism that forces supposedly random choices to collude. Or maybe there are very very rare events that will adjust the experimental results to be within Bell's bounds if we simply collected enough data for enough time. 
And I realize you weren't asking for certainty, just strong experimental evidence. What usually happens is you make a large parameter space of possible gravitational theories and based on your experimental evidence you start ruling out vast swaths of possible theories. And the fact that GR is still viable is reassuring. But other theories can fit the results too.
If we are wrong about something as basic as which things are close and which are far then there is lots of room to be wrong about a bunch of things. But stuff works pretty well.
And the fact that the universe acts so much like it were curved is enough of a thing to explain that it makes sense to study the theory.
Since there is always room to be wrong, you really look for principles, for reasons. We want to understand the universe. A curved space explains why different things move the same way under gravity, you say that spacetime is curved that way. If you want a different theory you should state a principled reason for a particular theory then we can see where the two theories are different and see if it is feasible to distinguish them experimentally. That's the good way. The bad way would be to be prejudiced against GR and just pick a random flat space theory that agrees with observations so far to the extent we've tested them so far.  It's bad because you picked it randomly so even if we rule out the section of parameter space it lives in you can just randomly pick another one all over again. No progress is made. And if instead you made a flat space theory that just makes perfectly the exact same predictions about experiments it isn't really a different theory it's just a reformulation and then you'd need a principle for your theory other than that the universe can look curved without really having to be.  Getting into a conversation about the way things really are without making different predictions can easily be physics adjacent rather than physics.
So to contrast again with Bell, long ago a minority thought that distant correlations would be different than quantum theory predicts. And they thought so for principled reasons. But the predictions of quantum theory held, as almost everyone expected. This took away the principle from the alternative. There are still hidden variable theories that are studied seriously but almost all hidden variable theories today are designed to agree with quantum predictions not to make different predictions. So they are like the reformulation of GR as a flat space theory.
And most physicists won't be interested. If the alternative formulations end up being easier to compute with, remember, teach, simulate, store, record, transmit or even if they make it easier to make parameter spaces of alternatives or easier to unify with other branches of physics there can be value. But those are the values of reformulations. Not of a principled alternative theory.
So while I haven't cited a no-go result hopefully you realize that productivity comes from either having a principled reason for an alternative that makes different predictions. Or else to make a mere reformulation you know the kinds of things that could make that valuable. And for that value they need to be valuable to people that also learn GR since if the universe acts like it is curved people will want to know what a curved universe acts like. So it has to be useful to people that also learn GR.
A: Too long for comment, but may suite as an answer.
As I understood the key is "The principle of equivalence of gravitational and inertial masses". Inertial mass defines how matter moves thru space-time and gravitational mass defines how it is affected by gravity field. While these masses are equivalent we can state that gravity defines space metric for moving matter.
If we would observe another universe where inertial mass is equivalent to electrical mass(charge) then it would be electrical field that defines space-time curvature.
To see it mathematically. If we throw stone under certain angle to horizon - its trajectory will be curved and we can calculate curvature based on motion equation using space-time metric. Curvature equation will depends on gravitational field only - not on initial condition how stone was thrown.
A: "Curvature" is just a word, and if you want to use all the math of general relativity but change the word to something else, you can do that. However, if you ask a mathematician to define curvature, they'll say that it's basically a measure of the path-dependence of parallel transport, and this is exactly what we observe, for all physical processes that can be described as parallel transport. Gravitational lensing is one example, but the gyroscopes aboard Gravity Probe B act according to the same rules.
You give a list of plausible-sounding Newtonian interpretations of various phenomena. Gravitational waves, however, are a phenomenon that you're never going to be able to shoehorn into such a description. Newtonian gravity is instantaneous action at a distance. Fundamentally we know that classical gravity has to be described by some kind of classical field theory, analogous to E&M, because special relativity forbids instantaneous action at a distance.
You can make a description of gravity as a spin-2 field on an unobservable background of flat spacetime. This is done in Deser, Gen Rel Grav 1 9(1970), http://arxiv.org/abs/gr-qc/0411023 , and there's a more readable presentation given in Misner, Thorne, and Wheeler, pp. 179ff and 424ff. Locally, this theory makes all the same predictions as GR. However, unlike GR, it's not compatible with any topology other than that of Minkowski space. So for example it can be used to describe an open cosmological model, but not a closed one.
