Many trees have evasive root systems but the rule of thumb is if you provide too frequent watering, roots will get lazy and stay near the surface. Try less frequent watering, but longer watering, or as they call it, "deep soaking." That will force the tree roots to aim downward, searching for the water.
If you value the tree...especially if the tree is still young...leave the root alone. An older tree can deal with a cut root. But lawns are over-rated anyway. Who is going to really care if the root is visible?