Turner was the son of James Turner, head of the Gurney and Turner's Yarmouth Bank and Elizabeth Cotman, the only daughter of the mayor of Yarmouth, John Cotman. He was educated at North Walsham Grammar School (now Paston College), Norfolk and at Barton Bendish as a pupil of the botanist Robert Forby. He then went up to Pembroke College, Cambridge, where he left without a degree due to his father's terminal illness. In 1796 he joined his father's bank and married Mary Palgrave, the daughter of William Palgrave.
He became interested in botany and published a number of books. In December 1802 he was elected a Fellow of the Royal Society.  In 1816, he was elected a foreign member of the Royal Swedish Academy of Sciences.
By 1820 his interest in botany had been replaced by an interest in antiquities. He and his children were taught drawing by renowned Norfolk artist John Sell Cotman who became a good friend. They travelled to Normandy together and collaborated on a book, Architectural Antiquities of Normandy, published in 1822, with Cotman providing the etchings.
By his first wife he was father-in-law of Sir William Jackson Hooker, FRS and of Sir Francis Palgrave, FRS and the grandfather of Sir Joseph Dalton Hooker, FRS and Sir Robert Harry Inglis Palgrave, FRS. After his first wife's death in 1850 he married Rosamund Duff and moved to live in Old Brompton. He died in 1858, and is buried in Brompton Cemetery, London.
- Synopsis of British Fuci 1802
- Muscologia Hibernicae Spicilegium (Irish Moss Ferns) 1804
- Botanist's Guide through England and Wales with Weston Dillwyn 1805
- Annals of Botany - nine articles 1800-1808
Dawson and Mary Palgrave had 11 children:
- Maria Dawson Turner (1797–1872), married William Jackson Hooker, botanist; their son was Joseph Dalton Hooker, also a botanist.
- Elizabeth Turner (1799–1852), married Francis Palgrave (né Cohen), historian, who took the name Palgrave upon conversion to Christianity.
- Dawson Turner (1801 – 1806)
- Mary Anne Turner (1803 – 1874)
- Harriet Turner (1806–1869), married John Gunn, clergyman and naturalist.
- Hannah Sarah Turner (1808 – )
- Dawson Turner (1809 – 1809)
- Katherine Turner (1810 – 1811)
- Eleanor Jane Turner (1811–1895), the youngest daughter, married William Jacobson, divine.
- Gurney Turner (1813 – 1848), whose son Dawson Turner played in the first international rugby match in 1871
- Dawson William Turner (1815-1885), educationalist.
- Lee, Sidney, ed. (1899). "Turner, Dawson". Dictionary of National Biography 57. London: Smith, Elder & Co.
- "Library and Archive Catalogue". Retrieved 29 October 2010.
- Stephen, Leslie, ed. (1887). "Cotman, John Sell". Dictionary of National Biography 12. London: Smith, Elder & Co. pp. 285–7.
- "Author Query for 'Turner'". International Plant Names Index.
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