What do women, gays, and other groups who generally support left wing politicians think of this betrayal? How can they ignore these injustices and keep supporting leftist pols?
They don't actually think at all, for the most part. Their conditioning is just too ingrained to be overcome and they live in the world of the true believer, even as they believe in completely contradictory concepts. The behavior strikes me as being almost cult-like in nature.
There is this British gay man I have encountered at another board that you just may have heard of, and his defense of any and all things Islamic is just as thorough as, say, Tommy Taint here. No matter how many times people point out the degree to which Muslims hate gays, and the extent they will go to follow through on this hatred, he just continues along spouting his dogma.
These are nothing more than useful idiots and they operate in a world of phobia. They may CALL others the phobic ones, but their entire political outlook consists of sacred cows that cannot be challenged. They are the new authoritarians who operate according to group think, and the one thing that terrifies them more than any other is to risk being ostracized from the group.
Voice ANY objection to Islam or the attitudes of Muslims, and the name calling begins. The fear of being called names motivates the extreme conformists into following suit. It doesn't have to make any sense. It only has to march in lock step.