Born in Hove near Brighton, Bevan studied at the Westminster School of Art before moving to Paris to study at the Académie Julian. He continued to live abroad afterwards, moving to Pont-Aven in Brittany in 1893. It was there that he met Paul Gauguin who was a great influence on the young artist.

After marrying the Polish painter Stanislawa de Karlowska he settled in London in 1900. He was introduced to the Fitzroy Street circle of artists by Spencer Gore and Harold Gilman who had noticed his work in the Allied Artists' Association exhibition of 1908 and went on to become a founding member of the Camden Town Group and the London Group, as well as being to the New English Art Club in 1922.
 
Bevan's most frequent subject was everyday street scenes, particularly horse-drawn cab yards and scenes of horse trading in London, but he also painted landscapes, in particular of Somerset and Devon. He first visited Devon in 1912 when, together with fellow painters Charles Ginner and Spencer Gore, he went to the Blackdown Hills to stay with a retired Argentinian rancher called Harold B Harrison. Harrison had enrolled in old age at the Slade School of Art and liked to invite artists whose work he admired to stay at his farmhouse, Applehayes in the village of Clayhidon. Harrison had studios erected for his visitors and Bevan returned in the summers of 1913 and 1915 to paint there. In 1916 Bevan took a cottage of his own in the Bolham Valley not far from Clayhidon, and in 1920 he moved to his own cottage in Luppitt in East Devon.