Dawson Turner was born and spent much of his life at Great Yarmouth in Norfolk. He was admitted as an undergraduate at Pembroke College, Cambridge, in 1793, but returned to Yarmouth before graduating, in order to take his place in the family banking business.
For some years Turner's chief interest was botany, particularly mosses, and he published several works on the subject and corresponded with many of the notable botanists of his day. In later life he concentrated on antiquarian pursuits, amassing a valuable collection of historical documents and autographs, as well as a substantial library which was eventually dispersed in a series of sales. He was a Fellow of various learned bodies, including the Royal Society, the Linnaean Society, and the Society of Antiquaries.
In 1796 Turner married Mary Palgrave, by whom he had eight surviving children. Mary Turner and her daughters were talented amateur artists; they were tutored in drawing by John Sell Cotman and also mastered the arts of etching and lithography. Between them they produced a significant number of sketches and prints, especially portraits and architectural studies, examples of which were often used by their father to embellish his books.