Curzon was the fifth son of Nathaniel Curzon, 1st Baron Scarsdale, and his wife Caroline. He first entered the Navy on 14 October 1776 as an able seaman aboard . When "Aeolus" returned from the West Indies in 1780, he became a midshipman aboard . "Magnanime" was sent out to assist in the Relief of Gibraltar in 1781, and then to the East Indies in 1782, to reinforce the fleet under Sir Edward Hughes. Curzon transferred aboard the flagship, , and fought in the battles of Providien, Negapatam, and Trincomalee. On 1 February 1783, he received a lieutenancy on "Monarca" and was with that ship at the Battle of Cuddalore in 1783, his last action in the East.