Once the ship has undocked and turned into the main shipping lane tugs would not normally be required.
The thick smoke indicates that full power was demanded from the engine. Think of the smoke from the stacks of a heavy truck powering up a hill.
Most of these ships have a large single propeller for efficiency.
With any propeller you have a torque or 'P' effect that will push the ship (or plane) sideways when high power settings are used.
The turn to the right was most probably the result of the emergency full astern power application.