Educated as a surgeon, L'Estrange began to practise on Chatham Street in Dublin around 1778.
On 12 June 1786, L'Estrange was appointed Assistant-Surgeon to the House of Industry Hospitals, and was for many years Surgeon to the Marine School.
He delivered the future poet Thomas Moore in Dublin on 28 May 1779.
L'Estrange married a Miss Spiels; their son, Francis, a Fellow of the College, became a dentist.
L'Estrange was appointed Justice of the Peace for Westmeath at a time when surgeons rarely held such a position.