Serialized fiction, whereby full-length novels were published in short, continuous sections either independently or in periodicals, was hugely popular during the 19th and early 20th centuries. The vogue for this approach was spearheaded by Charles Dickens, whose monthly Pickwick Papers (1836-7) were eagerly anticipated by a hooked public.
Many of Henry James’s novels were first published in serial form, with The Europeans (1878) and The Portrait of a Lady (1881) appearing in Atlantic Monthly, Washington Square (1880) in Cornhill Magazine, The Bostonians (1886) in Century Magazine and The Ambassadors (1903) in the North American Review. The Turn of the Screw was first published in weekly instalments in Collier's magazine between 27th January 1898 and 16th April 1899. An interesting essay on the Collier's version by Peter G. Beidler can be read here.
The first chapters of Washington Square in Cornhill Magazine: