It's unfortunate that your sister is sending you the message that she pretty much thinks the worst things possible. There is no refuting prejudice, the information is readily available for everyone to look up, and find out, and study. The article begins by question beginning a conclusion, by suggesting a prejudice, or preconceived opinion and perception is fact, and then proceeds to engage in confirmation bias, looking for any and all information that supports the prejudice, while delegitimizing, and ignoring information that contradicts the prejudice.
The fact that your sister gravitated to, and sifted through all of the information, and links that she came upon, until she found something like that, betrays her own need to confirm prejudice.
I'm not immune to aversion, and hate speech, I do find that highly distressing. What I read of it, began with a prejudice (have a gut reaction to these photos, and then agree that questioning that prejudice in anyway is ridiculous... moving on), then dismissed opposition as either weak minded, soft hearted, or in an evil conspiracy. Suggested that ->-bleeped-<- is an attack on women, and then named some names of purported transwomen that said things they believed and approved of, and therefore are the only honest, and truthful ones (their credibility presumably founded upon saying things I agree with).
The problem is that they're far more wrong here than merely a difference of opinion, or an ignorance to obscure facts, or something -- there is a heated antagonism, that is alarming, and distressing, and this is the cause of the opposition.