According to this and a few other websitea she copped to having had anorexia when she was younger. She's already pretty young so I don't know how young she would have been at the time. She did have a stint in the hospital for kidney failure when she was a child, the official cause was e.coli but who knows. Anyway, I don't know if this is legit or not, judge for yourselves.
http://capelino.com/lily-rose-depp-on-the-fight-against-anorexia-i-had-been-trying-to-get-better/
This is from an interview with Elle France in 2016. http://m.elle.fr/People/La-vie-des-...-ca-a-ete-dur-d-en-sortir-3330809#xtor=CS5-88
She talks about the accusations on Instagram about her being too skinny: "It saddens me because when I was younger, I struggled to recover. People with that eating disorder know how hard it is ... I've been trying to gain weight, and I'm proud of how far I've come."