I Am Curious
This movie was banned in Massachusetts because it was practically pornography. I Am Curious depicted many sexually explicit scenes, in which the protagonist was canonically under the age of consent. When this movie got banned by the governor of Massachusetts, the Supreme Court overturned the ban, and changed the textbook rules of what makes a movie "obscene."
Pink Flamingos
Do you know who John Waters is? You probably know his most prolific work: Hairspray. But, before Hairspray received major critical acclaim, Waters produced a movie called Pink Flamingos, which got banned in several small towns around America. The movie was banned because it depicted suggestive homosexual and heterosexual intercourse and indecent exposure. However, this movie's banned status actually made it significantly more famous within the communities that it was made for. So in a way, the ban was a double-edged sword.