Good news here is that the overwhelming majority of people don't go to the bathroom to harm others. Those who do are so few in number I'm not even sure they can be called a minority, just isolated incidents.
If you live in an area with a decent population, you might find a little comfort in the fact that someone in your family has likely already been in a bathroom with a transgendered person and they came away unharmed. With estimates of 6 in 1,000 identifying as transgendered, it's likely that your entire family has been in the bathroom at the same time a transgendered person was. Nobody was harmed, and I'd wager none of you was even aware of it.
Think of it this way. At any given Cowboys game there are (statistically) 600 transgendered people spread across the spectrum of transitioning from one sex to the other. At some point in time these people went to the bathroom and you can rest assured that at some point there were penises in the women's room and vaginas in the men's room. Nobody knew.