Exshaw trained under Dublin based architect and painter,
Francis Bindon. Bindon may have encouraged Exshaw to travel and study abroad. He may have won a medal for drawing while in
Paris, possibly from the
Académie Royale. In 1755, Exshaw returned to Dublin, holding a sale of the sculpture, drawings, and paintings he had collected in Europe on
Dame Street in the rooms of
Francesco Geminiani. He then travelled to
Rome in 1759, etching the model of Italian painter
Carol Maratta from life. He went on to visit
Amsterdam, where he especially studied the works of
Rembrandt, and executed two etchings from his pictures, "Potiphar's Wife making Accusation against Joseph" and "Christ with his Disciples at Sea in a Storm", the latter plate being dated 1760. He also executed some etchings and mezzotint engravings of heads of boors and peasants after various Dutch masters, and a mezzotint engraving of 'A Girl with a Basket of Cherries, and Two Boys,' after
Rubens. Returning to Dublin in 1762, he held 2 more sales on 10 February 1762 and in May 1764. He subsequently settled in
London in 1762, and unsuccessfully attempted to establish a drawing-school, after the example of
the Caracci, in Maiden Lane, Covent Garden. Exshaw was one of the early competitors for the Society of Arts' premium for an historical painting, with a picture of ‘The Black Prince entertaining the captive French Monarch after the Battle of Cressy.’ ==Death==