Having searched and found no definitive answer, here is my best guess.
In the Middle Ages Durham was important politically as it was a buffer state between England and Scotland. From 1075 (after the Norman Invasion) the Bishop of Durham became known as a Prince-Bishop, granted certain autonomous powers such as the right to raise an army, mint his own coins, and levy taxes, on condition he remained loyal to the English (ie Norman-French) king and fulfilled his role of protecting England’s northern frontier.
County comes from the Old French term, conté or cunté and could denote a
jurisdiction in mainland Europe, under the sovereignty of a count or a viscount. - ie, it was a word used by the Norman conquerors in England.
Shire on the other hand is an Old English word already given to many parts of England before the Norman Conquest.
Similarly, Ireland was never invaded by the Saxons but was conquered by the Normans. Thus the areas under direct French Norman rule - Ireland and Durham - were known as counties, while the shires of England retained their English shires.
You can find a parallel to this with food words - cows and sheep were the animals tended by the Anglo-Saxon peasants, beef and mutton were the meats eaten by the French nobility.
And the reason county goes before the name instead of after it is because the French speak backwards :)