Fan is turning slowly - Intertherm P3RC-048K Air Conditioner

There is a capacitor connected to the motor. It needs to be replaced.

If there is a pull chain on the fan, make sure it is on the high setting. Also you could check the capacitor to make sure it's not going bad. Otherwise you could need a new receiver.

Check your capacitor. If this is an add on remote, set the fan on high with the remote then pull the chain on the fan until you get high speed.

That's the radiator fan malfunctioning. Check the fan motor itself by conecting direct to battery while disconected from harness. If not OK replace. If OK check the fan switches, These are sensors that turn the fan on when temperature rises. If bad replace.

Because of the Voltage, many of the fans have relays. You can get free wiring and fusebox diagrams at Autozone.com after registering your vehicle. That is if you want to go back to stock.

I reread your statement and I think you mean that you tested the fan separately but when it is connected to the car it will only turn slowly.

Test the voltage going to the fan even though you did not use the stock connector. Look at the wiring diagrams and see if the power wire to the fan is controlled by a resistor or Relay which is limiting the fan speed. This is probably in the opposite direction of the connector you bypassed.and is meant to work automatically with a temperature sensor tripping it.

The fan bearings are probably gummed up with dirt or solidified lubricant. Unplug the unit and try turning the fan by hand. It should be easy to turn. If it is not check to see whether you can free the bearings with a quick rinse with WD-40. (If the bearings are sealed, you should replace the fan motor.)

Most systems work this way: temp sensor on engine sends a signal to the computer which in turn sends signal to blower relay that then sends signal to the fan.
Any problem along the way, and the fan won't come on.

it may have pulled in some hair around the shaft. disassemble and clean

If you've had it for a few years, the bearings are probably getting bad (dry).
Always start a fan on high---then go to the speed you want, only the high speed windings will handle start-up.