She also admitted three charges of causing him unnecessary suffering and two of administering him a poison - her own anti-depressant medication which had been put in his milk bottle.
Previously, the trial was delayed when Turtle was said to be feeling unwell.
Half a day was lost when she had apparently self harmed after buying a razor at the prison shop. She was said to have caused herself superficial injuries. The court previously heard that Turtle had said: "I wish I had never done it. I know I need help."
Turtle sat in the dock today with her head bowed, crying quietly. She is expected to be sentenced on Thursday.
The judge said he had read reports from forensic psychiatrists instructed by the prosecution and defence - and he would be able to take those reports into account.
Gordon Cole QC, defending, said the jury had not heard the psychiatric evidence but said there was relevant material available which would go to mitigation.
Infanticide was no longer an issue, but post-natal depression was, he said. While she had no mental illness it was agreed that she suffered from a personality disorder.
Mr Justice Lewis said there had to be a life sentence for murder, but he needed to fix the minimum term she will have to serve, and must consider was her culpability.
Turtle was remanded in custody in the meantime. During the trial, prosecuting barrister David Elias QC told the jury that after putting forward different versions of events, she accepted that she had caused his death but did not mean to do it.
Turtle accepted that she stopped him breathing on three occasions within a ten-day period. It was the third episode which led to his death.
On May 31, 2016, she deliberately pinched his nose so that he was unable to breathe and he turned blue. Paramedics were called, he received hospital treatment and he was discharged back to his family, apparently in good health.
But the following day, on June 3, when alone with her son, she deliberately suffocated him again when he became floppy. He was resuscitated, he suffered a fit on the way to hospital, and doctors could find no medical reason for his condition.
James returned home on June 6 but three days later the prosecution say she murdered him. Mr Elias said that she put her hand over nose and mouth and stopped him breathing despite the fact that he was struggling against her.
He was flailing his arms about and it went on for between five to ten minutes, he said.
She sat with his life-less body for several minutes, went to check on her mother-in-law, went to the toilet and, on her return to her room, pretended that she had just found him.
James did not recover from the last episode. He suffered brain damage from a devastating lack of oxygen and blood, and died in hospital on June 13 aged 58 days.
Turtle initially denied being responsible but then told a social worker and repeated it to a nurse that she had killed him.
She said she needed help and had heard voices telling her that she did not deserve James. "I wish I had never done it. I know I need help," she said.
The jury was told that when refused permission to go to the funeral she made false allegations against her partner Mr Hughes.
But in a later letter to the police she admitted that she was responsible. Turtle had denied murder, three charges of ill-treatment and two of administering poison.
