> Why are young girls so vulnerable to mental disease trends?
Nothing to do with young girls in particular, nor "mental disease" in particular.
Adolescents are constantly trying to find some axis along which they can excel/stand out, in a way that gives them societal power. (They're trying to "win", without really caring what it is they're winning at. They're just aware that right now they're not winning along any axis, and that has to change.)
For women specifically, being beautiful is one thing you can "win" at that can give you power.
Now, separately, anorexia can happen to anyone—recent research has suggested it's a sort of dual to obesity, another way the ventromedial hypothalamic nucleus (VMN) can become deregulated and cease to output the correct signal in response to fluctuations in leptin levels. For obesity, that makes people eat when they're "full" (because they don't feel full); for anorexia and bulemia, that makes people not-eat when they're "hungry" (because they don't feel hungry, and do feel disgusted/nauseous at the idea of eating more just like someone who's already stuffed full would.)
Like obesity, anorexia isn't necessarily genetic, but rather seems to be a disordered "state" that the brain can get into, which means that it potentially has environmental "triggers", even if it's purely a neurohormonal problem from that point forward. Some scientists hypothesize obesity as being the result of "re-training" your brain by forcibly exceeding your calorie budget. Anorexia is hypothesized similarly: "re-training" by forcibly starving yourself.
So: take teenage girls who want to "cultivate beauty" in order to gain social power. Some of them think they need to be thinner in order to be more beautiful. At this point, they're okay. After dieting, some are still okay. For others—perhaps the ones who saw the largest gains in social power from their loss of weight—something in their brain breaks a bit, and now they can't eat regular amounts of food any more without feeling sick.
Note that that could have happened to anyone, at any time, who decided to go on a diet. We don't understand exactly what makes people more "vulnerable" to anorexia, but adolescents aren't especially vulnerable. Teenage girls are just more likely to be people who are trying to diet for the first time, and therefore are more likely to be triggering and exposing an underlying vulnerability to anorexia for the first time—which we end up describing as "developing" anorexia.
Compare also: the guys who obsessively lift weights to gain muscle mass—because "cultivating your physique" is social power for teenage/college-aged boys—but then, once they start, their "unfitness" becomes a self-image problem, and they can never be happy with how they look, no matter how "ripped" they get.
Compare also: the development of Narcissistic Personality Disorder. People find that "cultivating a charismatic public image" gives them social power, but then become unable to stop optimizing their public image, even to the point that they habitually lie to their loved ones to present every action in the best possible light.
I feel like there's some unifying neurological principle to how these disorders develop, though I've never heard one suggested.