Really you want it through a relay. What current is your switch rated at? You need 30A minimum without a relay - and 2.5mm cable.

The fan leads are not +ve and negative as you descrive in your diagram. One is ignition live (black) and the other switched live to the fan.

To make your setup work - disconnect the wire you have put to earth - and connect it to the other fan plug wire - so that the switch bridges the two connections.