Are both the low and high speed control signals connected to the high speed relay? They need to be. The low and high temp switches are exclusive. When the low switch is closed, the high switch is off and vice versa.
A better way to do this that doesn't require fiddling with the car's wiring is to eliminate the fan speed resistor or bridge across it.
A better way to do this that doesn't require fiddling with the car's wiring is to eliminate the fan speed resistor or bridge across it.

Comment