The design of the mounted portion of this daily quest is so bad that I have to assume it is bugged. You cannot control the tauralus you’re riding in any way; instead, all you can do is spam buttons on cooldown and hope that there isn’t an opposite faction player in front of you stealing tag credit on mobs.
Seriously, the path this thing rides is atrocious. You have to have intended for players to fully control this once it reaches the main area.