Glinsky was eight years old when he came to America with his parents to join relatives in Syracuse, New York. He studied art at Syracuse University and, beginning in 1916, in New York at the Cooper Union, the City College of New York, the Beaux-Arts Institute of Design, and Columbia University. At the start of his career he was associated with architect Albert Kahn, in Detroit, Michigan, creating architectural sculpturs in stone and metal. In the last years of the 1920s Glinsky was working in Italy and France, and in 1929, in Paris, his first one-man show was presented.
Among Glinsky's most visible achievements are his bronze doors for the French Building at 45th Street and Fifth Avenue in New York, and The Waters of Life Fountain at the All Faith's Memorial Tower in Paramus, New Jersey. He modeled busts of Wilbur Wright for New York University's Hall of Fame, and of Eleanor Roosevelt for the United States Labor Department, Washington, D. C. He also executed busts of artists Harriet Frishmuth and Leon Kroll, and of historian Will Durant. During the 1930s he worked extensively under the auspices of the Federal Art Projects.
Glinsky taught at the Beaux-Arts Institute of Design, 1931-32, and 1940-41; Brooklyn College of the City University of New York, 1949-55; and at Columbia University, 1957-61. From 1950 to his death, he was an adjunct professor in the New York University School of Continuing Education.
Glinsky first exhibited in an Academy annual in 1921, and received the Academy's Speyer Prize in 1970. Among his other honors was a Guggenheim Fellowship awarded in 1930. He was a founding member and at one time executive secretary of the Sculptors Guild, and also a member of the Architectural League of New York, and its vice president 1956-58. The sculptor Cleo Hartwig and Glinsky married in 1951.
