What experiments show that special relativity is valid in the absence of gravity?

The phrase "special relativity is invalid in presence of gravity" is sufficiently misleading, on its own, to be a useless statement. Rather we should consider the size of the region of space and time in which we want to discuss a physical phenomenon, and compare this size to the spacetime curvature effects associated with gravity. Special relativity is valid in the limit where the size of the region (whether in distance or time) is small compared to the local radii of curvature of spacetime, and the experimental effects are measured relative to a frame in freefall.

There are three strategies you can adopt to test special relativity near to a large body such as planet Earth.

1. First strategy: do your experiments in a horizontal plane. For example, you can test time dilation in particle accelerators and in atomic physics experiments. (Such a plane is not in freefall, but the horizonal component of its motion is the same as that of a frame in freefall, namely zero).

2. Second strategy: do your experiment in freefall.

3. Third strategy: do your experiment in any setting, but take the gravitational effects into account. (For this one you would be testing general and special relativity at the same time.)

The second strategy works very well in satellites for example. What happens is that freefall motion cancels out any effect of gravity which is simply proportional to the acceleration, and then only the remaining effects are relevant. So even though Earth's gravity is there, the prediction from general relativity is that in free fall the theory of special relativity should apply. So it can be tested. For the highest precision you need to check the effects you are looking at are not sensitive to the variation in Earth's gravity from one place to another. The famous observation of time dilation and space contraction for muons moving through the atmosphere is an experiment of this type. The muons are in free fall. Earth's gravity does vary between the upper atmosphere and the surface of the Earth, but not by much, so you can use special relativity to good approximation (and it turns out that in fact the corrections introduced by G.R. to the muon observation are really tiny).