The “supper club” started in London and became popular in the second half of the 19th century. A “supper club” is a “night club,” with food, music and dancing. An 1898 newspaper article (below) stated that New York City then had only one “supper club.”
 
The “supper club” fad took place in 1913, when dances such as the “turkey trot” exploded in popularity. Murray’s Roman Gardens, at 228 West 42nd Street in New York, was the leader in this “supper club” craze. The introduction of jazz in 1917 enlivened the supper clubs, but many closed in 1920 with Prohibition. Some large supper clubs existed in New York in the 1920s and 1930s, but they became more rare by 1940—killed off by World War II and other forms of entertainment.
 
   
Merriam-Webster Online Dictionary
Main Entry: supper club
Function: noun
Date: 1925
: nightclub
   
(Oxford English Dictionary)
supper club, a restaurant serving suppers and usu. providing entertainment;
1927 H. CRANE Let. 19 Dec. (1965) 313, I don’t think I’ll dare attend that *supper club again.
1977 New Yorker 8 Aug. 66/3 Only after a time of heading a hotel and supper-club trio did he turn his hand to composing and arranging.
   
Google Books
The Club and the Drawing-Room:
Being pictures of modern life: social, political, and professional
By Cecil Hay
London: R. Hardwicke
1870
Pg. 25:   
There are pre-eminently dining-clubs and supper-clubs, and their corresponding
varieties of coffee-houses.
(London—ed.)
     
20 October 1889, Galveston (TX) Daily News, “Some Midnight Clubs,” pg. 20, col. 3:
He also owns a club—the Supper Club—one of the oldest and best known establishments of its kind in the city of London.
 
The Supper club is in Percy street, just off Tottenham Court road, in the center of the Bloomsbury district, where 50 per cent of the boarding and apartment houses of the metropolis are situated. Like the Spooferies, the SUpper club caters to the social and recreational requirements of ladies and gentlemen whose artistic talents render it difficult for them to find entertainment elsewhere after their hours of work; but need it be said that its patrons comprise an overwhelming percentage of gentlemen who never passed the stage door of a theater, and ladies whose other sources of income never necessitated their donning the tights and spangles of burlesque.
 
Google Books
Paris
By Émile Zola
Paris: Bibliothèque-Charpentier
1898
Pg. 158:   
It’s not the young fellows of dinner and supper clubs, who issue manifestoes and indulge in all sorts of extravagance.
     
6 June 1898, Kansas City (MO) Star, “New York’s One Supper Club,” pg. 2:
From the New York Sun
New York has had but one supper club and that solitary imitation of a peculiar London fashion had an unfortunate and unappreciated existence. It was conducted on a plan somewhat different from that prevailing in the home of these establishments. They are commonly supposed to be a necessity in London, because the restaurants close at so early an hour and the (illegible—ed.) which they meet does not exist here in a degree likely to bring about any further attempts to establish one here. Even in London they are not unmolested. It was under embarrassing circumstances to the proprietor that this description of one of the best known of these places was made known to the public.
 
Google Books
The Night Side of London
By Robert Machray
London: John Macqueen
1902
Pg. 171:
Then in addition to the dinner-clubs there are the supper-clubs, of which the most fashionable and popular is the Grafton.
   
Google Books
The Man of Pleasure
By Ralph Nevill
London: Chatto & Windus
1912
Pg. 41:   
The fact is, supper clubs in private rooms are not popular with the very people — ladies — for whose benefit they are intended.
     
15 January 1913, Kansas City (MO) Star, pg. 16:
SOCIETY TROTS IN SECRET
A NEW YORK CLIQUE BARS ALL EX-
CEPT MOST EXTREME DANCES.
Nothing Tamer Than the Bunny Hug Is
Tolerated in the Supper Club, Made
Up of Some of the Lights
of the Social Circle.
Karl K. Kitchen in the New York World.
If you want to stay up all night you’ve got to have a woman with you. If you haven’t you’re barred from the newest, livest and gayest social organization in New York.
 
I refer to the Supper Club.
 
Have you heard of the Supper Club? probably you haven’t, for it was only organized last month. It is the very last word in clubdom.
 
No member or guest shall be admitted to the clubrooms at any time unless accompanied by a woman. There must be at least one woman to every two gentlemen admitted to the clubrooms.
 
This is the most important house rule of the Supper Club. For let it be known that this club was organized for the propagation of turkey trotting in all its divers and sundry ramifications.
 
IS OPEN EVERY NIGHT.
Seven nights a week its clubrooms, which embrace the entire third floor of Murray’s on West Forty-second Street, are open to its four hundred members from dinner time until 4 o’clock in the morning. In case the numbers do not care to leave at that hour a quorum of the house committee can suspend the rules and the turkey trot, the Maxie, the Tango, the horse trot, the bunny hug, the chicken trot an the one-step can be danced until 8 o’clock in the morning. To keep the clubrooms open after 8 o’clock requires extraordinary action on the part of the baord of governors.
 
The membership of this unique organization—the only other organization in the world like it is the Grafton Gallery Club in London—is composed of the younger dancing set, who have transformed five of New York’s widely known restaurants into turkey trotting haunts. Such well known young men as Vincent Astorand Reginald Vanderbilt are members of the club, and all of its officers, with one exception, are to be found in the social register. A son of a former President of the United States is one of them.
 
10 April 1913, Boston (MA) Journal, pg. 5:
New York Society Folk Object
To 1 A. M. Closing of Restaurants
New York, APril 9.—The Supper Club, and organization of society men and women who meet nightly at an uptown restaurant, announced today through its attorney, James O. Cleary, that it would ask the courts for an injunction prohibiting the police from closing it up at 1 o’clock every morning, in compliance with Mayor Gaynor’s ukase.
 
The Supper Club does not open its dining room till 11 P. M. Its members have been in the habit of dining and dnacing till 2 and 3 o’clock.
 
Google Books
The night side of Europe as seen by a Broadwayite abroad
By Karl Kingsley Kitchen
Cleveland, OH: The David Gibson Company
1914
Pg. 183 (London): 
“You want to visit the supper clubs, do you?” asked my new acquaintance.
 
Google Books
A Land-Girl’s Love Story
By Berta Ruck
New York, NY: Dodd, Mead
1919
Pg. 207:
They go on to boxes at theatres, supper clubs.
   
Google Books
The Restaurants of New York
By George S. Chappell
New York, NY: Greenberg, Inc.
1925
Pg. 107:
THE SUPPER CLUBS: FROM AFTER THE THEATER ON