It is worth noting that the theory of Special Relativity is confirmed by the Michelson–Morley experiment, which happened 25 years before Einstein came up with his explanation. In the interim period, Lorentz had managed to come up with all of the contractions in Einstein's theory based on experimental data, but had no explanation of why it was true.
Einstein's insight was to come up with a single coherent theory that explained everything which had been observed, all together.
Therefore there was no period before they managed to confirm it.
The general theory was produced in 1916, when people were preoccupied by WW I. Its first confirmation was the precession of Mercury's orbit, which had been known about since the 1800s. Most people only heard about it after the second confirmation in 1919 when Eddington measured gravitational lensing during an eclipse. The third confirmation was the measurement of gravitational redshift.
Therefore it too was considered confirmed by the time most heard about it.
I say "considered" because several decades later physicists realized that all three confirmations are explained by a wide variety of plausible theories that do not possess the nonlinear characteristics of general relativity. (Removing nonlinearity would be desirable because it would make it easier to reconcile gravity and quantum mechanics.) This drove the development of better confirmations.
Note that theories are usually developed to explain existing observation, so the presence of confirmation before explanatory theory is actually quite common.
I may well be misremrembering a comment in a class that I took over 20 years ago.
Wikipedia lists a number of alternatives to general relativity. But I can't find any that look linear. There may not have been a linear theory, just a set of heuristics that any linear theory would be likely to support. In particular when I took general relativity, my professor made the following points.
Let's model a concrete photon coming out of a gravitational potential well. We can calculate exactly how much mass it should lose. Apply the particle wave duality to that energy loss we get GR's gravitational redshift to first order. Based on the wave interpretation, you can predict, to first order, the amount that time slows in a gravitational well.
To model Mercury, assume that the Sun creates a potential well. At all points in its orbit, assume that Mercury moves as Special Relativity says that a body with fixed angular momentum and the appropriate kinetic energy for that radius should move. (I forget whether you need to toss in the aforementioned time dilation.) To first order, you recover the GR correction to the precession of Mercury.
I forget the argument for the bending of light (maybe just throw gravitational time dilation into the mix?), but there is a first-order heuristic that can give the correct figure there as well.
The point is that if a series of heuristic arguments pull out the correct prediction for the original 3 tests, then any theory that tries to reasonably combine QM, special relativity, and gravity, is likely to give similar predictions to first order.
That professor claimed that the first test which he did not have a heuristic explanation for involved the actual time that light took to get from one point to another. Looking on Wikipedia, I think he's talking about http://en.wikipedia.org/wiki/Shapiro_delay.
Einstein's insight was to come up with a single coherent theory that explained everything which had been observed, all together.
Therefore there was no period before they managed to confirm it.
The general theory was produced in 1916, when people were preoccupied by WW I. Its first confirmation was the precession of Mercury's orbit, which had been known about since the 1800s. Most people only heard about it after the second confirmation in 1919 when Eddington measured gravitational lensing during an eclipse. The third confirmation was the measurement of gravitational redshift.
Therefore it too was considered confirmed by the time most heard about it.
I say "considered" because several decades later physicists realized that all three confirmations are explained by a wide variety of plausible theories that do not possess the nonlinear characteristics of general relativity. (Removing nonlinearity would be desirable because it would make it easier to reconcile gravity and quantum mechanics.) This drove the development of better confirmations.
Note that theories are usually developed to explain existing observation, so the presence of confirmation before explanatory theory is actually quite common.