Norman began performing as a stand-up comedian in strip joints and nightclubs while producing his first Off-Broadway play, the New York premiere of John Arden's Live Like Pigs,[3] which opened on June 7, 1965.

His performance in the title role of Bertolt Brecht's The Resistible Rise of Arturo Ui at the Stamford Center for the Arts in Connecticut (1980) was locally acclaimed.

[9] Over the course of his career, Norman would go on to act in and produce scores of movies, raising in excess of $100,000,000 for motion picture production, most notably with French producer Henry Lange, with whom he made over a dozen films[10]— including the 1971 vampire lesbian cult hit, Daughters of Darkness—and with Bert Schneider: Hearts And Minds (Warner Bros., 1974), The Gentleman Tramp (1976), and Paramount's 1977 Tracks, directed by Henry Jaglom who would become Norman's most frequent moviemaking partner.

[20] In 1998, Norman acquired the catalog of the American Play Company (founded 1889) for himself and actor-producer Michael Douglas for their newly formed joint venture, the American Entertainment Holding Company (AEHC),[21] which controlled the rights to thousands of plays and manuscripts by such authors as John Steinbeck, Tennessee Williams, George Bernard Shaw, Eugene O'Neill, Thornton Wilder, Oscar Wilde, A.A. Milne, and Maurine Dallas Watkins.

[22][23] In 2006, Variety's Elizabeth Guider wrote of Norman: "There are people through the decades who become regular fixtures in the pages of Variety—everyone from Al Jolson to Jimmy Durante to Michael Ovitz to Harvey Weinstein.

He made his television debut in 1953 at the age of twelve on WBZ-TV Boston's Community Auditions talent show as a drummer with his band, Howie Zuker and His Music Makers.

Subsequently he guest-starred in such popular series as The A-Team (1985) and Baywatch (1993), had a recurring role on The Nanny (1993–1995)[31] and was featured in several TV movies including At Home with the Webbers (1993).

[34] Norman was an art collector, counting among his acquisitions five pieces by Jean-Michel Basquiat, whose “Untitled” sold in May 2017 at Sotheby's New York for $110.5 million, setting the then record price for an American artist at auction.