'Turk' in reference books
J.S. Farmer & W.E. Henley, Slang and Its Analogues (1904) has a fairly lengthy entry for turk:
TURK, subs. (old).—1. a sword [other old slang terms for "sword" cited by Farmer & Henley include andrew, fox, and toledo (or tol)]. 1638. Albino and Bellama, 108. That he forthwith unsheath'd his trusty turke, Cald forth that blood which in his veines did lurk.
2. (old).—A savage fellow ; 'a cruel, hard-hearted man' (B. E. and Grose); a TARTAR (q.v.). Also, TO TURN TURK=to turn renegade, to change for the worse, to GO OFF (q.v.). ... In modern usage Turk has lost somewhat of its rigorous meaning, and is frequently employed as a half-jesting endearment to mischievous, destructive boy : e.g., 'You young Turk!" [Citations omitted.]
3. (old).—A target : a dummy made up of cloth and rags.
Perhaps the most interesting thing about this entry—aside from its mention of the usage of "young Turk" just four years before the Young Turk Revolution in Turkey—is its silence with regard to any connection between turk and Irish Americans.
Also free of any reference to Irish people is Joseph Wright, The English Dialect Dictionary (1905):
TURK, sb. ... 1, A violent, savage man ; a cruel taskmaster. [Citations omitted.] 2. A tiresome, mischievous child. [Citations omitted.] 3. Used as an intensive for anything big or formidable of its kind. [Citations omitted.]
Harold Wentworth & Stuart Flexner, Dictionary of American Slang (1960), however, does note the association of turk with people of Irish ethnicity:
Turk [or] turk n. 1 A strong man; a large, strong, energetic, overbearing man; a man quickly aroused to anger; a stubborn man, one hard to deal with. Orig. applied mainly to and used by the Irish and people of Irish descent. Now fairly common; often a nickname given to a prize fighter. From the Gaelic "torc" = a wild boar. ...
On the other hand, Eric Partridge, A Dictionary of Slang and Unconventional English, fifth edition (1961) asserts that turks in connection with the Irish is a feature of (fairly recent) Teddy boy slang:
Turks. The Irish : Teddy boys' [slang] : since ca. 1949. (The Observer, March 1, 1959.)
Three pitchers named Turk
When I was young, one of the stars of my home-town baseball team, the Houston Colt .45s, was a right-handed starting pitcher from Boston named Dick Farrell, who was known as a fun-loving, hard-partying person, but with a somewhat hot temper. His nickname was Turk—and according to an article about him posted on the Society for American Baseball Research website, his father was known as Big Turk. I don't know whether the father's Irish ethnicity had anything to do with that nickname, but he may have been Big Turk by the 1930s (Dick Farrell was born in 1934).
Aside from that rather tenuous connection, I have never heard an Irish or Irish-American person referred to as "Turk" or "a turk," and the usage may be obsolete or nearly so in the United States. Another pitcher, though from a generation later, Turk Wendell (born in Pittsfield, Massachusetts in 1967), is said (in his SABR biography) to have acquired his nickname as follows:
Turk got his nickname at the age of 3. By one account, the little boy repeatedly jumped face first from a window into a mound of snow made by his grandfather. The man said, “That was something only a Turk would do.” In 1991 Wendell himself clarified: “My grandfather nicknamed me after one of his buddies because I was always doing stupid, rebellious things.” “He wrecked everything in no time at all,” said his father. “He always thought he was indestructible. He always has been a daredevil.”
A third baseball pitcher, Turk Lown, who was born in Brooklyn, New York, in 1924, "got his nickname as a child because of his fondness for eating turkey," according to a Wikipedia article about him. Neither Wendell nor Lown is an Irish surname.
The most likely sources for the term turk as applied to an Irish or Irish-American person, I think, are Turk (a native of Turkey) as a byword for someone savage (an expression of bigotry that was probably several centuries in the making), and torc (the intemperate and unmanageable Gaelic wild boar). I seriously doubt that turkey (the North American bird) has anything to do with the usage.
UPDATE (2/3/2017): Early newspaper mentions of 'Irish Turks'
The earliest instance that I've been able to find of Turk as a direct insult term (that is, as a fighting word) for an Irish person is from "Can He Be Trusted with a Club: Policeman McCullough Gets into a Quarrel with Two Women," in the [New York] Sun (July 2, 1888):
Policeman Patrick McCullough of the Charles street station charged Mary Walsh of West Houston and Hudson streets at Jefferson Market Court yesterday morning with disorderly conduct In Hudson street on Saturday night, and with calling him a "big Irish Turk," He also accused Bernard Cahill and Francis Reilly, young neighbors of Mrs. Walsh, of interfering when he arrested her.
Mrs. Walsh accused McCullough of insulting her daughter, Mrs. Louisa Stanberry, and of clubbing her when she rebuked him. Mrs. Stanberry said she called the policeman's attention to the fact that two men had beaten a third in front of her home. He did not find the assailants, and, returning, he said to her:
"You little tar, you ought to mind your own business."
Her husband and mother spoke up for her and the policeman hit her mother and knocked her down.
Cahill said that McCullough had his club lifted to strike Mrs. Walsh, when he caught hold of the club and prevented the blow. Then McCullough clubbed him. His head was done up in bandages.
Reilly said that he went to Cahill's rescue.
Mrs. Walsh said that McCullough used her so roughly that he broke a diamond ring from her finger, and she showed the broken band. McCullough's story was that "the gang tried to do him up," and that Mrs. Walsh fell down. Justice Gorman discharged Mrs. Walsh and Cahill and fined Reilly $5. Mrs. Walsh paid his fine. McCullough was tried by the Commissioners lately for beating a citizen.
This story is interesting, in part, because the surnames of those mentioned suggest that everyone involved may have been of Irish descent.
More-ambiguous instances of "Irish Turk" in newspaper accounts go back to the 1840s. From "City Intelligence," in the New York Herald (June 9, 1842):
A BIGAMIST WITH THREE WIVES AND THREE ALIASES—A man whose real name, he says, is Patrick Casgran, of 74 Allen street, was committed at the Upper Police yesterday, charged by Michael McCarran, of 26 James street, with the crime of bigamy. On the 13th of November, 1841, he was married by the name of Patrick McClusker to Julia Henson, the sister of his deceased wife, by the Rev. James McDough, of St. James's Church, in this city. Taking a notion into his head, as he says, that the marriage with his wife's sister was illegal, on account of relationship, he left her and on the 26th of February, 1842, was married to Mary Ann Lorman by the Rev. M. A. Stillwell, under the name of Patrick McCasgrove alias McCasker. These circumstances being confirmed by the affidavit of Mr. McCarran, the Turk as locked up for trial.
If the decision of the late Presbyterian Synod of Philadelphia is correct "that it is incest for a man to marry the sister of his deceased wife," the defence of the Irish Turk will stand good before a jury.
Here it appears that Turk is being used as a slang term for "bigamist" and that Irish Turk is simply a juxtaposition of the defendant's ethnicity with his presumed marital status. But "Irish Turk" appears in a somewhat similar context 22 years later. From "Broadway Below the Sidewalk," in the New York Clipper (April 23, 1864):
Everybody has read of old Bluebeard with his plurality of wives, and most people are opposed to his summary method of doing things. Bluebeard was an Irish Turk, and not a Turkish Turk, which may account for his not over eleemosynary propensities. But Bluebeard's style wouldn't suit Gotham, because our folks are proverbial for the kindly manner with which they treat the gentler sex on all occasions, and woman-beaters have no more show than the elder Law had when he ran for President.
Other nineteenth-century instances involve music-hall entertainments that seemingly focus on Irish characters. From an advertisement in the New York Clipper (September 10, 1881):
MARCHING SONGS.—“The Killarney Musketeers,” “The Petticoat Brigade,” SONGS AND DANCES—“Rose perfumed Bouquet,” “The Two Irish Turks,” “Where the Sweetest Flowers Grow,” “Where Pretty Lilacs Grow,” “Pretty Little Lou,” “Sunny Days,” “Little Roguish Eyes,” “Pretty Little Dandelion,” “Pretty Irish Queen,” “Sweeter Than a Red Ripe Peach.” EMERALD ISLE GEMS—“Bright Little Spot on the Ocean,” “Erin’s Green Isle,” “Sons of Erin’s Isle,” “A Dear Spot of Land,” “Picking up Gold in the Street.” ... 27c. each. Any 4 for S1. Lieder’s Publishing house, 61 Chatham Street, N. Y.
From "Thespian Temples," in the Leadville [Colorado] Daily Herald (January 8, 1882):
Paddy Hughes will introduce one of his best specialties, entitled, Dremni Drhue and the Irish Turk. Mr. Hughes is a first-class artist, and a genial favorite.
From an advertisement for the Alhambra Music Hall, in the Sydney [New South Wales] Morning Herald (August 20, 1888):
Serio Comic—"It's English you know"—Miss LETTIE LWYNE and TA TA TRA LA LA LA.
The Australian Team, DELOWERY, GRAYDON, and HOLLAND as the IRISH TURKS.
Comic—"Right before the Missus" and "Captain, Call the Mate"—Mr. HARRY HASTINGS.
To conclude with the Cottiers' Laughable Farce, THE AMERICANS IN PARIS.
Delowery, Graydon, and Holland continued to perform this act for more than a decade, as we see from an advertisement in the Perth Western Australian from February 12, 1898, promoting "DELOHERY. CRAYDON, and HOLLANE In one of their funniest and best Irish character sketches, 'Irish Turks.'"
The association of "Turk" with Irish people goes back well into the nineteenth century—with "big Irish Turk" unmistakably intended as an insult in an incident reported in 1888. The very early occurrences (from 1842 and 1864) of "Irish Turk" in connection with a person who has multiple wives is intriguing, but not unmistakably a set phrase. Instances of "Irish Turks" as stock characters in popular entertainments occur in both the United States and Australia by 1888.