Turnbull was born in Edinburgh and grew up in the North of England and in Canada. He studied Natural Science at Cambridge University and graduated in Medicine from the University of Pennsylvania in 1951.
In 1957, Turnbull started Migrant Press, one of the first British-run presses to focus on poets in the modernist tradition. His own books include A Gathering of Poems 1950-1980 (1983) and Rattle of Scree: Poems (1997). He now lives in Edinburgh again and works on what he terms kinetic poems; texts for instalation in which the movement of the reader and/or of the text becomes part of the reading experience.