Of American-English origin, the phrase cold turkey denotes a method of treating a drug addict by sudden and complete withdrawal of the drug, instead of by a gradual process.

This phrase alludes to the goose pimples, resembling the skin of a cold turkey, that a person experiences as a side effect of the treatment.

These are the earliest occurrences of the phrase cold turkey that I have found, in chronological order:

1-: From The Evening Bulletin (Providence, Rhode Island) of Friday 12^th January 1917:

CHARLES A. WISE, DRUG ADDICT, DIES AT JAIL HOSPITAL
Young Man Sentenced Week Ago in Federal Court Unable to Stand Effects of Being Cut Off from His Accustomed Supply of Narcotics

Charles A. Wise, a drug addict, who was sentenced to four months in the Providence County Jail a week ago, died at 12:05 this noon. Death was said at the jail hospital to have been due to morphinism, otherwise morphine poisoning. His death is the second from the effects of drugs here in two weeks.
Wise had been cut off absolutely from any use of the drug since being committed to jail. He had been addicted to drugs for nearly eight years, he said in court, and was accustomed to take from 40 to 60 grains a day.
Before being taken to jail, Wise expressed terror of the treatment he knew he would receive there.
“Do you know what they’ll do to me?” he asked his custodians while waiting for sentence. “They’ll give me the ‘cold turkey’ treatment—just strap me down to the bed without letting me have any dope, and then let me suffer.”
Dr. Henry A. Jones, Superintendent of the almshouse, said this afternoon that Wise’s condition had been bad since coming to the jail. This morning he was attended by Dr. Fred Works, interne at the hospital, who found him steadily sinking. Dr. Jones said that Wise was of a highly nervous temperament, and that he had also suffered from alcoholism and that these two factors, together with his addiction to morphine, had brought him to a precarious physical condition.
[…]
Several addicts who have been treated at the jail hospital have recovered under the care they received there. Aaron Hutchins and Grace Adams, both colored, who appeared before the last Federal grand jury, looked hardly like the same persons when brought into court after some time spent at the institution. Neither had meanwhile been allowed any drugs.
Dr. Charles O’Leary, who was recently sentenced to three months in jail, was said to-day to be doing very well under the treatment for the drug habit.

2-: From The Tulsa Daily World (Tulsa, Oklahoma) of Sunday 16^th March 1919—in the following confession, a drug addict named Harry C. Gilson explains that, when he arrived at Tulsa, Dr. Robert E. Lee Rhodes persuaded him to go into the county jail and “take the “cold turkey” cure”:

Confessions of a “Dope Fiend”
Drug Soaked Wreck Comes to Tulsa From New York—Finds Sympathy and Takes the “Cold Turkey” Cure in Tulsa County Jail on Advice of County Physician Rhodes—Out of Jail After Three Weeks Without the Craving for Drugs and With a Determination to Be a Useful Citizen—Describes the Fight He Made to Overcome Dope.

[…]
I don’t care what anybody says, the best cure is “cold turkey.” What I am referring to is to put a man in jail and if he isn’t man enough to use a little will power along with a great deal of sufferings then he will have to let the cold prison steel act as his will power. I’ve suffered the agonies of hell in the county jail here and I was looking for a rope to hang myself with for the first three nights. The first 72 hours is terrible for a man who goes to jail to break his habit. After that he becomes very hungry and very little sleep does he get. I didn’t sleep in three weeks in the jail.

3-: From The Sun (New York City, N.Y.) of Wednesday 30^th July 1919:

PELHAM HOSPITAL WILL COST $90,000
Saving After Drug Clinic Closes Will Cut This Estimate.
READY IN FORTNIGHT
Dr. Robert J. Wilson to Be in Charge at City Sanatorium.

Dr. Royal S. Copeland, Health Commissioner, began yesterday in real earnest to formulate plans to take over for the treatment of drug addicts the abandoned hospital buildings formerly used in connection with the work of the naval training station at Pelham Bay Park. […]
[…]
Asked whether the Worth street clinic’s work will be continued after the Pelham Bay hospitals are in operation, Dr. Copeland said: “My judgment is that the clinic on Worth street will be through with its work in from two to three months. This place was opened to meet the emergency of caring for the addicts who were unable to get their prescriptions from the private practitioners, arrested under the Harrison law by the Federal authorities. These patients either will be off the drug in three months or else they will be at the hospital. The function of the emergency clinic will then have ceased.”
Reports from several hospitals relative to the cases of addicts who had been sent from the clinic to be cured, nearly all of them giving assurances of speedy cures, failed to impress Dr. Copeland.
“I am not satisfied with the methods in vogue in this city at present for curing drug addicts,” he said in referring to this matter. “It has been customary for hospitals to give what the addicts call the ‘cold turkey treatment,’ which means that the patient is taken off the drug at once. Two or three days later the addict is declared to have been cured.
“I have a report from one hospital at which seven patients are said to have been ‘cured’ in two days, and another from one where eleven are reported as ‘cured’ in two days. I feel confident that this practice will be met and corrected through the opening of the new hospital, where individual treatment will be given to each addict.”

4-: From The Boston Herald (Boston, Massachusetts) of Sunday 27^th March 1921—Dr. Harry H. Colburn was the physician at Charles-street Jail:

HOW DRUG ADDICTS GET THAT WAY
BEST CURE IS NECESSARILY HARSH
They Live a Life of Unreality, Are Distinctly Below Normal and Their Trouble Is Mental Rather Than Physical
By John E. Pember

[…]
Dr. Colburn’s practice is to stop short the supply of drugs to all prisoners who come under his jurisdiction. […]
The plan of cutting off the supply at once aroused considerable criticism and prisoners, thus deprived of their daily “shot,” became almost uncontrollable. But the doctor stuck to it and maintains that results have justified his action.
“Cold turkey,” as the addicts call it, is a heroic remedy, but so far as the doctor’s knowledge goes it has never resulted fatally, and after the discomfort of the first few days of deprivation has passed most of the victims themselves come to agree that his method is most effective and satisfactory.
“We give milk and sometimes strichnia [sic] to sustain the heart action,” said Dr. Colburn. “No one ever knew a ‘hoppie’ * to die of cold turkey. Some are completely broken of the habit in this manner, and when their sentences expire leave the jail cured, as far as we can judge.
“But in others the craving is too deeply implanted. There is a physiological change which makes the drug seemingly essential. Whenever opportunity is offered these addicts return to the practice and are soon in as bad shape as ever.
“When the drug users are committed to jail and their supplies are cut off short they are very restless and uncomfortable at first. They feel pretty miserable for several days or longer, dependent on how confirmed they are in the habit.
“They cannot sleep, and walk up and down their cells incessantly. They can neither lie nor sit still for more than a few minutes at a time. Their muscles twitch uncontrollably.
“Catarrhal symptoms develop. A characteristic sneeze is a particularly marked symptom. Sometimes they suffer from cramps and intestinal troubles. One stage is known as the ‘chuck horrors.’ There is a great demand for candy and sweet things. The system seems to demand sugar.
“The addict during this time is not an agreeable object. Far from it. But after six or seven days he begins to act like a rational human being again.
“The whole condition in addicts seems to depend on the strength of the heart. With a weak heart they soon drop off. They would not have time to become what we call addicts.
“But a man with a strong heart to begin with seems to be able to live forever. He can survive physiological storms and stresses that would put the ordinary clean-living, normally constituted man out of business in short order, even if it did not have fatal results. He has immunized by repeated doses.”

* This is the definition of hoppie, from the above-quoted article published in The Boston Herald (Boston, Massachusetts) of Sunday 27^th March 1921:

HOPPIE. An addict, doubtless derived from “hop,” the more or less familiar Chinese word for opium.