The Humans are for all practical purposes babies at intergalactic warfare and presence, while the Turians are an ancient spacefaring race with a strict military meritocracy.
They invested in large quantities of Plot Armor.
In practice, the humans did not fight the Turian Hierarchy to a standstill. The entire human fleet and combined military fought a single turian scouting flotilla to the tune of about six hundred casualties on each side.
Calling it a war is, frankly, a gross exaggeration.