Conybeare was born at Pinhoe, where his father was vicar, and educated at Blundell's School and Exeter College, Oxford. He was elected a Probationary Fellow of Exeter College in 1710, took his B.A. degree in 1713 and was appointed a year later as Praelector in Philosophy.
On May 27, 1716, Conybeare was ordained as a priest by the Bishop of Winchester, Sir Jonathan Trelawney and took a curacy in Surrey. He returned to Oxford a year later and became a well known preacher.
His subsequent appointments included:
- Curate of St. Clement's, Oxford
- Senior Proctor, Exeter College, Oxford, 1725
- Elected Rector of Exeter College,1730
- Dean of Christ Church, Oxford, 1732
- Bishop of Bristol, 1750
Conybeare was known for the publication of his book Calumny Refuted, an answer to the personal slander of Dr. Richard Newton.
- Defence of Revealed Religion, 1732
- Calumny Refuted, Published for J.J. and P. Knapton [etc.] (London), 1735
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