Bastard Out of Carolina, based on the childhood of writer Dorothy Allison, contains harrowing scenes of the abuse and rape of a 12-year-old girl by her stepfather. Mr Turner funded the $5m film, which was to be shown by his Atlanta-based TNT network, but after seeing it he said it was "inappropriate" for the late-afternoon slot which had been set aside.
Senior executives employed by his company had already made major cuts to the film, which stars Jennifer Jason Leigh as the mother of the victim, who is played by 12-year-old Jena Malone.
Instead, TNT is now allowing Miss Huston, the 44-year-old daughter of the legendary Oscar-winning director John Huston and herself an Oscar- winner thanks to her performance as a Mafia moll in Prizzi's Honour, the right to sell her work world-wide.
The film should have been screened this spring in America but it will now be shown in cinemas, at least in Europe, later this year.
The film has been attracting attention at the Cannes Film Festival, where it is in competition for the Camera d' Or award. Certain sequences, however, were so shocking that some women in the audience at screenings felt obliged to leave the auditorium. They were visibly upset by scenes in which Ms Malone is molested and horrifically raped by her stepfather, played by American actor Ron Eldard.
Actress Jane Fonda, who is married to Mr Turner, is said to have screamed aloud when she witnessed the attacks at an earlier private screening.
In Cannes yesterday Ms Huston, who is married to the artist Robert Graham, was attempting to play down the bitter dispute over the film. She said: "I was disappointed initially, but I have to say that Mr Turner was perfectly within his rights to censor the film. Contractually, I had no rights in that area whatsoever.
"Having said that, I am glad it is going to be seen and that is what matters. There were many cuts made to the TV version but I still felt privileged to have been given such a free rein. Friends told me that my father gave this advice to young film makers: 'Swing the bat'. That's exactly what I did. I attempted to swing the bat."
The film, set against the backdrop of a poor community living in rural South Carolina in the Fifties, follows the sad fortunes of illegitimate Bone Boatwright through her childhood days. Her mother eventually leaves her for her stepfather, even with the knowledge that he has been sexually abusing her on a regular basis.
Bone's character is based on the unhappy childhood of the writer Dorothy Allison. Though she acted as a consultant on the film, she refused to be employed as a voice-over narrator because, said Ms Huston: "She did not want to go back to that time and place."
Ms Huston said she was "filled with trepidation" over the most numbing scenes of violence, in which the child was seen being assaulted with a leather strap.
The young actress was chaperoned on the set of the film by her mother but Ms Huston said: "More than anything I was concerned with the mental and physical safety of Jena.
"For the molestation scene it was merely a case of doing one rehearsal. As for the rape, I put them together with a stunt man who was very protective towards her. It was like a dance which had been fully choreographed with the understanding that if she felt uncomfortable she would be able to stop and express herself.
"This kind of story should be told. This kind of man could be a neighbour from down the street or the guy next door. That's how I feel.
"Child abuse is a germ, a fungus, a cancer and a disease."
Her encounter with censorship has not deterred Ms Huston from considering other directing offers. She said: "I don't know if I'd want to shoot another story as grim and as exacting as this. I just hope this film makes people think about how they treat others."
Other stars in the film include Lyle Lovett, the ex-husband of Julia Roberts, young Hollywood star Christina Ricci and Diana Scarwid. They all agreed to take a fraction of their normal wage to work with Ms Huston.