People used to think that lobotomies could heal such diseases like depression, schizophrenia and even homosexuality. You need to use "could" here.

"People used to think that prefrontal lobotomies could cure depression, schizophrenia, and even homosexuality." It's not necessary to explain that depression and schizophrenia are mental illnesses. "Cure" is the idiomatic word here, at least in US English. I think it's best to use the full phrase "prefrontal lobotomy."

People used to think that lobotomies could heal such diseases as depression, schizophrenia and even homosexuality. SUCH usually is followed by AS, not LIKE.