Bjarnarson was born in Laufás near Eyjafjörður, the son of the Reverend Björns Halldórsson and his wife Sigríður Einarsdóttir.

On 19 March 1885 he was appointed pastor of Akureyri; however, on 28 August he became a lecturer at the Prestaskólar (Seminary) and later its director in 1894.

On 19 September 1908 he was appointed Bishop of Iceland to succeed the retiring Hallgrímur Sveinsson.

They had four children, including Tryggvi Þórhallsson who became Prime Minister of Iceland in 1927.

His other three children were Svava, Björn and Dóra who married Ásgeir Ásgeirsson and who became President of Iceland in 1952.