Well, there have been no experiments where we take a meter stick, accelerate it to near light speed, and then measure the length. That's not feasible with our technology (or with any foreseeable in the near future). However, that's certainly not the only way to test Lorentz contraction. Indeed, the original Michelson-Morley and Kennedy-Thorndike experiments require both time dilation and Lorentz contraction to explain the results. There are other ways to measure contraction as well. For example, we can't directly measure the length of heavy ions moving at relativistic speeds as they zip through particle accelerators. However, from relativity we'd expect them to be shaped something like pancakes rather than little spheres from the perspective of the lab frame. And indeed, if you try to treat them as spheres, you get the wrong scattering results, but if you apply Lorentz contraction and treat them as compressed in the direction of motion, you get results that match observation.
Relativity makes a very long list of predictions, some of which involve time dilation alone, some of which involve Lorentz contraction alone, and some of which involve both. Some of those predictions cannot be tested easily, but many of them have been. I stand by my statement that Lorentz contraction has been clearly shown by experiment many times in the century since Einstein introduced relativity.




