Recep Cetin, appearing in court alongside his alleged accomplice father, Erup Cetin, appealed for the media to be banned because of the effect reports were having on his mental health.
He complained he was being treated like, "a Christian, an atheist or communist" in the trial.
"I am a Muslim. My father is innocent, yet he has been in prison for one and a half years. Camera records show my father was not at the scene," he added.
Cetin claimed he was suffering from depression because of media coverage of the case and called for the court to remove reporters from the courtroom.
"I have read reports claiming I cut off the victim's heads," he said.
"I murdered them, that is true. But the media has reported false allegations."
Despite his admission of guilt, Cetin defended his father and denied that he had any involvement in the double murder.
"I swear to this country, to this nation, that whatever my father says is true. I am guilty and my guilt is a sufficient punishment for me," he said.
Cetin's father told the court he was the victim of a conspiracy. "I have nothing to do with this incident. I am innocent yet I am kept in prison. Neither God nor justice will forgive this," he said.
The women's bodies were found in woodland near their holiday resort, in Kusadasi, Turkey. Both were with Ms Graham's daughter Shannon (16), who had started a relationship with Recep Cetin.
Ms Graham and Ms Dinsmore had hoped to convince Cetin to stop attempting to marry Shannon. They got in a car with Cetin, who said he would take them shopping. Prosecutors allege Cetin drove them to woodlands, stabbed Ms Graham in the chest and cut Ms Dinsmore's throat.
Police arrested Recep Cetin and the waiter confessed to the killing. He is believed to be 22-years-old, however he insists he is 17 and should be tried as a child.
Cetin is currently being tried alongside his father in an adult court.
The case was adjourned until March 20.