2004 Jeep Liberty Overheating

There is only a few components that can cause the vehicle engine to overheat. I will list some components and there purpose.
Reservoir pressure cap: This serves two purposes, one to maintain cooling system pressure (about 16lbs of pressure at operating temperature) and second, cause a vacuum to within the system to suck more coolant from the reservoir into the cooling system or press to push out coolant from the cooling system into the reservoir.

Thermostat: Regulates cooling system operating temperature by opening and closing a valve to prevent coolant flow throughout the cooling system.

Water pump: Pumps coolant through the cooling system (when the thermostat opens) creates flow and is always pumping or cycling coolant.

Heat exchangers or more commonly radiator or heater core: These give off heat to the air that goes through the radiator or heater core.

Coolant: a 50/50 mix of coolant and water is required. If you have just straight concentrated coolant it can cause heat exchange issues because water retains heat so much better and releases it.

If you replaced the thermostat and added a 50/50 ratio of coolant and water (or 70/40 of coolant and water if you live in cold climate such at Alaska) and there is no air in the cooling system. Then the only other issues could be the water pump is not pumping coolant or the air fins on the radiator are restricted or the radiator is restricted or the cooling fans are no coming on.
Worst case could be a blown head gasket or cracked cylinder head or engine block pushing exhaust gases into the cooling system causing overheating while over pressurizing the cooling system usually forcing coolant out of the reservoir.
I hope this helps. When does the overheating occur? At freeway speed or around town or idling in stop & go traffic?

First, you need to make sure the cooling fan is coming on. Try turning on the a/c (if it works) and look to see if the cooling fan turned on. If the a/c doesn't work, warm up the engine and see if the cooling fans comes on (take care not to overheat engine).

If this works, then touch (be careful almost everything will be hot under the hood at operating temperature) both the upper radiator hose and lower. If the water pump is working you should feel similar temperatures between the two hoses. If the pump is not working there will be a difference between the hoses but make sure there has been sufficient time to allow the thermostat to open up allowing coolant to flow through.

Last you will need to have the cooling system tested for combustion gases which your probably better off leaving that to a professional. Also, if there is a leak causing combustion gases to enter the cooling system it will cost quite a bit to have it repaired. So that being said, take care not to allow the engine to over heat because it will cause damage.
One thought came to me. Could it be possible that the thermostat was installed upside down (spring side on the thermostat facing up in the housing)?
Try this and get back to me.