Wrangler jeep heater/AC fan

The blower motor relay is energized when the Blower Motor Control is set to the High position. This removes the blower motor resistors from the circuit and battery voltage is then applied directly from the HVAC BLO fuse (RH IP Accessory Wiring Junction Block) to the blower motor though the relay switch contacts. The blower motor then runs at high speed.

HVAC Blower Controls Circuit Description
The blower motor is a variable speed motor. The motor operates at a higher rate when voltage is increased to the maximum level.
The blower motor resistors reduce the voltage supplied from the A/C FAN fuse (LH IP Accessory Wiring Junction Block) when the blower control is set at low and medium speeds (speeds 1-4). The blower motor relay is energized when the Blower Motor Control is set to the High position. This removes the blower motor resistors from the circuit and battery voltage is then applied directly from the HVAC BLO fuse (RH IP Accessory Wiring Junction Block) to the blower motor though the relay switch contacts. The blower motor then runs at high speed.

Hi Maria , The A/C system wouldn't keep the fans from running but none working cooling fans will keep the A/C from working ! This is a fairly complex system if your not familiar with this sort of thing ! I will add the description from the factory service manual.
Cooling Fan Control - Two Fan System
The engine cooling fan system consists of 2 electrical cooling fans and 3 fan relays. The relays are arranged in a series/parallel configuration that allows the powertrain control module (PCM) to operate both fans together at low or high speeds. The cooling fans and fan relays receive battery positive voltage from the underhood fuse block.
During low speed operation, the PCM supplies the ground path for the low speed fan relay through the low speed cooling fan relay control circuit. This energizes the low speed fan relay coil, closes the relay contacts, and supplies battery positive voltage from the low fan fuse through the cooling fan motor supply voltage circuit to the left cooling fan. The ground path for the left cooling fan is through the cooling fan s/p relay and the right cooling fan. The result is a series circuit with both fans running at low speed.
During high speed operation the PCM supplies the ground path for the low speed fan relay through the low speed cooling fan relay control circuit. After a 3 second delay, the PCM supplies a ground path for the high speed fan relay and the cooling fan s/p relay through the high speed cooling fan relay control circuit. This energizes the cooling fan s/p relay coil, closes the relay contacts, and provides a ground path for the left cooling fan. At the same time the high speed fan relay coil is energized closing the relay contacts and provides battery positive voltage from the high fan fuse on the cooling fan motor supply voltage circuit to the right cooling fan. During high speed fan operation, both engine cooling fans have there own ground path. The result is a parallel circuit with both fans running at high speed.
Do you have a check engine light lit ? The thing to do would have it check at the dealer or a ASE certified repair facility ! They could hook up a advanced scan tool an diagnose fairly quick ! Or you could find a wiring diagram an try testing with a volt meter yourself ! But watch yourself so you don't short out the PCM - engine computer !

I would check the high speed blower relay.

Replace the blower motor resistance it controls the fan speeds.

You'll have to find the correct wiring diagram in order to trace down the fault in your system.. Once you obtain the wiring diagram you'll be able to see what else beside wire is between the #4 speed setting and the blower motor. My guess is you might find a high speed fan relay is triggered from the #4 fan switch setting, applying full battery voltage to the blower motor. If locating the appropriate diagram proves to be difficult you can try looking around the underhood and passenger compartment fuse panel to find a relay. Keep in mind not all relays are always in fuse boxes which makes the diagram important.

This is 100% the Blower (Fan) Motor Resistor located near the Blower Motor.
This is a very common problem with Chyrsler, Dodge, Jeep and Plymouth. The key symptom is the Fan only working on High.
This is very easy to access and replace.

Let me know if this helped....

Yes I would go with the switch, BUT, does it have a resistor bank? That is, how does it reduce the speed? Many systems use resistors to lower the speed. Burnt or bad connections can cause this as well.
You can test the blower motor by connecting direct power from a source like the cigarette lighter if you are handy.
Most of the time the lower speeds konk out but the high speed continues to work this is the resistor bank that goes out.

blower motor resister has burnt out also check wiring going to your blower motor resistor these must have a good tight connection or they will burn if any sighns of being hot change the complete wiring pig tail to your blower resistor

I'd check the relay and the resistor ...don't have a location for yours but the resistor is always installed into the heater box so it stays cool. you can usually follow the wiring back to the relay as it's generally nearby. Resistor looks like a piece of circuit board material with several wires connected to it with two screws holding it to the heater box so it sits flush except for the wires. It has several coils inside...any that are broken have failed.

if fan speed is working only on one speed it is most likely the blower resister,located in the blower box