There are various manufacturers of wood lathes Intorex, Locatelli that turn bats. They have square steadies and round steadies. I have succesfully turned bats on a Hapfo and Intorex with no steady. The right cutter is imperitive and it often doesn't create good results without the right wood. I always place the handle at the headstock for rigidity. Just by looking I can tell your piece is whipping from severe vibration. Contrary to metal wood will often vibrate worse with a light cut than a heavy cut. A heavy cut puts the wood fibers under tension and keeps vibration to a minimum while a light cut allows the different grain densities to bounce the piece. A typical blank has face grain and edge grain both with distinct cutting characteristics. What wood are you turning? Typical bats are Ash or Hard maple preferably with a tight grain pattern typical of slow growth higher density lumber. I have turned Beech due to its cosistant grain. If you follow the cutter recommendations experiment with depths of cut, speed, and feed The last resort is a simple board say a 3' 2" by 4" with one end resting on the crosslide or in the bed between the ways applying pressure on the piece to absorb excess vibration. But I believe you will be successful once you find that magic combination of cutter geometry, depth of cut, speed and feed and machining sequence with the right material selection. Carry on young man trial and error is the foundation of success.
If it is true a person learns from their mistakes then I must be the smartest man alive.