At the 1973 World Snooker Championship, his first tournament on the professional snooker circuit, Thorburn defeated Dennis Taylor 9–8 in the first round then lost 15–16 to Williams in the second round. Later that year, he had a 4–0 win over Pat Houlihan at the 1973 Norwich Union Open before losing 2–4 to Higgins in the quarter-final. In the 1974 World Snooker Championship he defeated Alex McDonald 8–3 in qualifying then lost 4–8 to Paddy Morgan in the first round. He started the 1974–75 snooker season with a victory in the 1974 Canadian Open, knocking out Willie Thorne and Graham Miles to reach the final, where he won 8–6 against Taylor. He reached the quarter-finals of the 1975 World Snooker Championship with wins over Morgan and Miles, losing the quarter-final 12–19 to Eddie Charlton, and, the following year, was eliminated 14–15 by Higgins in the first round of the 1976 World Snooker Championship.