What is the spec of the 12V supply?
What is the rating of the desktop fan?
Is the supply powerful enough to drive the fan?
Bear in mind breadboard connections are good to 200mA to 500mA or so, put too much current
through and you'll ruin it - high current wiring best done with screw connector blocks which are
rated 5A, 10A etc.
BTW the common ground point is ideally at the motor driver since thats where signal levels meet high-current
circuitry.