He was last Bishop of Beverley and the first Bishop of Leeds. He entered Street Cuthbert"s College, Ushaw on 9 May 1830, and received the Tonsure and the four minor orders from Bishop Francis George Mostyn on 5 June 1841. During his last year at Ushaw, Cornthwaite taught Humanities.
He entered the English College, Rome on 30 September 1842, and took the oath there on 2 July 1842.
He was ordained a subdeacon in December 1843, a deacon on 3 March 1844, and a priest on 9 November 1845. After leaving the English College on 13 April 1846, he joined the mission at Carlisle.
He returned to Rome on his appointment as the Rector of the English College, Rome on 25 August 1851, remaining in that post until resigned in September 1857. On his return to England, he became the Missionary Rector of Street Augustine"s, Darlington and Secretary to Doctor William Hogarth, Bishop of Hexham and Newcastle.
He was appointed Cameriere d"onore extra Urbem (Honorary Prelate) to His Holiness on 16 July 1858, and made Canon and Theologian of the Chapter of the Diocese of Hexham and Newcastle.
Monsignor Cornthwaite was appointed the bishop of the Diocese of Beverley on 3 September 1861. His consecration to the Episcopate took place on 10 November 1861, the principal consecrator was Cardinal Nicholas Wiseman, Archbishop of Westminster, with Bishop Thomas Grant of Southwark and Bishop Richard Roskell of Nottingham serving as co-consecrators. On the 20 December 1878, the Diocese of Beverley was suppressed and it was replaced by the dioceses of Leeds and Middlesbrough.
Bishop Cornthwaite continued to serve as the Bishop of Leeds until his death on 16 June 1890, aged 72.
He is buried at the Church of Mary Immaculate, Sicklinghall.