I've often wondered why we're more nervous in the ladies' room than just about anywhere else. It's much the same experience as any other. If you're read, you're read. That's always uncomfortable, wherever it happens. The difference is that the consequences of being read in the ladies' room are potentially more serious than elsewhere. That's where we can get called "pervert", "sexual predator", etc., and that's where perhaps the management or cops can be called in. Not something to look forward to.
Nonetheless, I still think it's basically the same experience we have in lots of other places. We shop, we work, we go places for entertainment, etc. We've got lots of things to do, and whatever we're doing, there's always a risk that things won't turn out well for us.
My experience in the ladies' might be relevant. I've been using the ladies' for over two years now and have never once been challenged. I have the advantage of being very passable, and at 5'3" I don't call attention to myself. Also, my voice is very passable. It never gives me away.
Even so, I can be quite nervous at times. E.g., if there's a queue. You'd like to get in and get out, but if you're forced to wait, you fear the delay will give others a chance to read you.
Now a lot of people say that women are more likely to have a chat in the loo than men are, and so the chances are greater that you're going to have to engage with someone face-to-face (and voice-to-voice). I'd agree that on the whole women chat more in the loo than men--but it's usually when they're in a group. My feeling is that an individual woman isn't any more likely to engage an individual in conversation than a man is. This is to say that in two years I've very rarely been forced to get into conversation with someone.
Other people will no doubt disagree with me, but I personally haven't found the ladies' all that different from the gents'. Individual women tend to just go about their business and then leave--which gives you the chance to do the same.
So my view is that the risks we face in the ladies' are often over-stated. I personally don't feel they're much greater than they are anywhere else. So my strategy in the loo is the same as anywhere else: I simply go about my business with confidence as if I have every right to be there (which I do). If you act like a woman who knows exactly what she's doing, I think the chances are that's what people will see and they won't challenge you because they'll hardly even take note of you.
The scariest experience I ever had: I was at a play and during the intermission I had to go. Knowing that there would be a rush, I got there in a hurry in order to get in and out before the place was too busy. Now it turned out that I was the first one in. But when I came out of the stall, the place was absolutely packed. I was terrified. But I simply reminded myself, "Be cool! Act like you're a woman who has every right to be here." That's what I did, and I didn't come to grief.
So my feeling is that even girls who are less passable will probably be OK in the ladies' as long as you don't do stuff that will attract attention to you. Perhaps I'm wrong about this. I hope not, because like everybody else on this planet sometimes we need to go--and like everybody else we should be able to exercise the simple right to go. And what makes me angry is this view that transwomen are a threat to other women. Rubbish! One of the main reasons to use the ladies' is because there are certain cisguys who are a huge threat to us.
At any rate, those are my views--but I'm well aware that other people's experiences will be different from mine, so their views will be different from mine.