Are we talking transexuals? The noise I've heard recently (including a TED talk by Ivan Coyote, BBC.co.uk etc) is from those identifying as "transgender" which (according to the Intersex Society of North America), 'are usually people who are born with typical male or female anatomies but feel as though they’ve been born into the “wrong body.”'
That's a whole different, er, ball game.
The "prove it" angle I suspect comes from the issue of transgender rather than transexuals. I'd be questioning the State's fulfilment of its duty of care in schools if a group of the (male) class clowns at my daughters' school decided to put on a bit of eye-liner, declare themselves transgender and had legal rights to use the girls' locker rooms, no questions legally allowed to be asked. Then you've got the issue of parent help. What happens if some forty-year-old male parent declares themselves to be transgender and wants to help supervise the fourteen-year-old girls getting changed for swimming lessons? The birth-certificate thing is never meant to be used, its meant to be a threat to prevent abuse of the system.
I'm sorry if that you feel you've been born that way. If you are one of the tiny fraction of people who consider themselves transgender, I'm really sorry but you should really be able to see how this is open to abuse and why it shouldn't be allowed. I don't know if its a "modern" thing, a "youth" thing, an American thing or a transgender thing, but some people find it hard to grasp that it isn't always about you.
There's more that needs to be noted with regard to the legislation in the US. Just because it is supported by some very unpleasant and bigoted people, that doesn't necessarily make it a bad piece of legislation. Just because it is opposed by some very successful and/or popular people doesn't make it good legislation. Bruce Springsteen's concert cancellation, Paypal's "reconsideration of its business plans" and all the others are amazingly hypocritical. They are exercising the the very right, (to not conduct business based on an ethical stance) that the legislation they oppose seeks to safeguard.
No-one cares who you want to sleep with or if your brain doesn't align with your body. No-one is denying you access to toilets, no-one is forcing you to use second-rate toilets, but if you deliberately walk into toilets designated for people with different primary genitalia, we are no longer talking about what you are, but about what you do. That is not a civil rights issue.
Slightly OT: The TED talk was interesting. The story was told of a little four-year old tomboy who wet her pants in prep and "learned the lesson that there was no safe place for her to pee." It was told with great humour, sadness and empathy. It is also completely irrelevant to the discussions in the adult world. Playing with tonka trucks and wearing camouflage doesn't make you transgender. Being a tomboy doesn't make you transgender. Going into the wrong toilets isn't uncommon for really young children. Kids laughing at that and generally being mean to each other is normal, even if it isn't nice. But you know what? Kids don't hold grudges and have "issues" like adults. They will often be mean to each other and then ten minutes later play together quite happily. Stop projecting adult issues onto children. The whole talk was a superbly executed piece of propaganda but it basically amounted to "Look! Kittens! If you disagree with me, you're responsible for killing these cute kittens." That is not a good basis for legislation.