Crosby was born in Stockton and after serving some time in the office of Benjamin Hoskins, a Sunderland solicitor, went to London. He made his way in city politics and in 1770 became Lord Mayor of London. Crosby's courageous struggle against parliament for the liberties of the press led to his being committed to the Tower in 1771. Eventually he was released and returned to the Mansion House in a triumphal procession. The upshot of Crosby's activities was that there has never been another attempt to restrain publication of parliamentary debates. Crosby was the cousin of Sir Cuthbert Sharp (q.v.)