I'm not asking you to agree with the other side, just to give it a fair shake before putting it down.
Consider the viewpoint of a parent of a young (10-year-old) child. That parent sends their 10-year-old daughter into the women's restroom. If that restroom can be legally entered by any person who is willing to say out loud "I feel like I am a woman", that restroom is open to any number of adult men who prey on children to enter.
Yes, this is the ever-present, sometimes-alarmist "who will think of the children" argument.
But being a tired argument doesn't make it an inaccurate or illegitimate one. If there is no answer given for it, then it remains valid, if eventually tiring to hear.
That's a conservative talking point, not the reality of the situation. In reality, a trans person will have to go far beyond just feeling like a woman to get away with entering a women's room; they will also have to take substantial steps towards looking like a woman.
You don't even have to feel like a woman to enter a woman's room, you just have to say you identify as a woman.
"Substantial steps towards looking like a woman." What does a woman look like? A woman looks like whatever she chooses to look like. Actually, she sort of looks like me after Taco Tuesdays where I have to use the restroom really badly and all the stalls in the men's bathroom are full.
If you are demanding that a "real woman" make herself subservient to what society thinks a real woman should look like, or maybe even get a note from a doctor diagnosing gender dysmorphia, I would advise you to check your cis privilege. /s