No, I think people on your side are illiberal because they are trying to change the law and take away rights women have fought for in the past, while preventing them from even meeting to talk about what the impact might be, pointing out likely harms or unfairness, or advocating for the interests of cis women (and in a few cases men) when they are in conflict with those of trans people. (I don’t know if you personally support this or not.)
There is no live debate on white supremacy or whatever other comparisons you might make - or if there is, it’s because certain people want to bring it back, and I fully support allowing anyone to speak in opposition to that!
Additionally, the idea that only members of X group get to decide what counts as prejudice against them is an illiberal one, and blatantly open to abuse. (I’m hoping some of the more dubious responses to the war in Gaza will help people realise that.) They should have a major input, but not carte blanche to declare any and all opposition to be prejudice.
Earlier you asked about negative results from left-wing policies. I was thinking about the 100,000,000 people killed by communism, and the sometimes dire results of electing socialist parties around the world. But Prohibition was supported by progressives, and that didn’t turn out too well. Eugenics was also supported by prominent progressives before the Second World War, while traditionalists like the Catholic Church opposed it. It’s harder to say definitively that recent progressive policies are having bad results, but there are plenty of plausible candidates, especially in California (imagine governing a state undergoing a tech boom, one so rich that the economy recently surpassed Japan’s, so badly that it actually loses population!) where they have been dumbing down education, failing to enforce laws against shoplifting, and allowing parts of major cities to be taken over by homeless encampments (despite spending $24 billion on homelessness initiatives over the past five years).
There’s a recent example from the UK of how fear of the consequences of speaking out about contentious topics can lead to harm, in the form of the grooming gangs scandal, as detailed here:
ETA:
I didn’t think this needed pointing out, but maybe it does: I’m opposed to censorship and cancellations on a wide range of issues, as I’m sure other posters can confirm. It’s very far from the case that I single out prejudice against trans and queer people as being okay. Nor do I support right-wing censorship or cancellation campaigns. I believe current society does not do enough to support free speech, and is far too censorious in general.