I have to admit that I feel poorly equipped to address this subject. I'm more interested in the feedback of others than in trying to persuade others of my opinions. I should also point out that although this topic is specifically about Special Relativity, much of what is said could also apply to General Relativity.
Special Relativity has had a profound effect on both science and modern culture. I was a young witness when I first started to learn about relativity. This wasn't exactly encouraged by other witnesses. However, I looked up relativity in the Watchtower index and found a few short articles discussing the basic concepts.
Most witnesses seemed uninterested in what I was learning and were reluctant to talk about it. Then I ran across my first official skeptic of Special Relativity.
We made an appointment to discuss it. I prepared my arguments for why I thought it was a valid theory. I considered the possibility that he would make the claim that it somehow contradicted the Bible. In that event, I was prepared to show him the Awake articles that presented relativity in a favorable light.
Our actual discussion turned out to be a big disappointment. He just said that he didn't think men could figure out and understand things like that. I tried to explain some of the evidence for Special Relativity, but he just didn't have any interest in hearing about it.
I got the impression that it was a little bit beyond what he could easily understand, so he was happy to discount it as silliness.
Later, I learned there was an entire subculture of people who disagree with Special Relativity. I became intrigued by the psychology of those who would put a lot of effort into trying to undermine a theory that appears to have strong experimental confirmation.
I assumed that the critics generally lacked either the imagination or education to properly understand relativity. My own reading of some of their writings confirmed that was sometimes the case. Many of the arguments are hopelessly flawed.
I could also begin to sympathise with some of their frustration. The scientific community is reluctant to address criticisms of Relativity. Critical papers have a hard time getting reviewed and letters to scientists often don't get a reply. Part of the problem is that scientists don't have the time to respond to sheer number of criticisms, especially since many of them have already been addressed when Relativity was young.
It still has to be frustrating. It's understandable how some would get the impression that the scientific community is overly defensive or hiding something.
There is also the factor that physicists have not been able to unify Quantum Mechanics with Relativity. This is a major stumbling block to further progress. It may be necessary to drastically alter our understanding of one or both theories in order to get past this hurdle.
It seems ironic that the scientific community is trying to avoid debates about Relativity when it might be necessary to question past assumptions in order to keep making progress.
Advocates of Relativity have also been guilty of overstating the evidence, explaining the theory improperly and making the same kinds of logical errors as their critics.
You will often hear it said that Special Relativity has been proven experimentally beyond a reasonable doubt. There has indeed been considerable experimental evidence to support certain aspects of Special Relativity. Although the interpretation of the experimental evidence is sometimes disputed, much of it does seem to be consistent with the concepts of time dilation and the constancy of the speed of light.
The concepts of length contraction and relativity of simultaneity have not yet been verified experimentally. Yet advocates of Relativity will often point to confirmation of one aspect of Relativity as confirming the entire theory, even though there are other theories that are also consistent with the available experimental evidence (such as Edwards' Theory and Lorentz Ether Theory).
I'm curious to see if others here have thought about or researched these issues. I have seen discussions on these topics turn nasty in other places, so I hope we can keep this discussion civil (but not so civil that this topic gets ignored completely).