He started as a clerk in a bank. His lifetime affliction with rheumatic fever began during his military service during the First World War. He entered the British Museum in 1921 where he worked for Charles Tate Regan (1878-1943).
From 1939 to 1944, he was in charge of the Natural History Museum at Tring as the Curator of Zoology.
Norman was the author of, among others, A History of Fishes (1931) and A Draft Synopsis of the Orders, Families and Genera of Recent Fishes (1957). He was considered closer to Albert Günther (1830-1914) than to Regan.