How to trubleshoot to overheating 2003 jeep - 2003 Jeep Liberty

no idea to be honest earlier jeeps had then at the front of the wing down below as i remember ?? newer one would i imagine be in the electrical box under the bonnet near the battery

if no air in engine block water jackets, did you purge it by the book?
check compression on all cylinders.,
at wide open throttle.
expect over 150psi or motor is toast.
and btw
all things can be fixed , there are are no limits.... ever.

Is it actually overheating?

It overheated.
You'll need to figure out if it is a faulty thermostat, water pump, cooling fan, or other causes.

sounds like coolant and that vehicle overheated or now has a leak ---and replacing plugs usually doesnt correct an overheating problem and if told had bad thermostat [ which will cause overheating ] i would have system tested for leaks then look into thermostat as a problem to your overheating problem

its either the temperature sending unit or a fan relay there is two of each

CODE P1512 - MANIFOLD DIFFERENTIAL PRESSURE CIRCUIT SIGNAL LOW.

electric fans or belt driven? if electric are they kicking on?

pickup a repair manual

Your Jeep should have a 192 or 195 degree thermostat. Do u have to add coolant once in a while? If there is no coolant loss, no overheating problem, u shouldn't have to worry about it, it could be the temp. sending unit is not accurate. If it concerns u it would probably be worth it to remove the thermostat, check the temp rating on it(it should be stamped on it). Replace w/ a new one, and see what it indicates for temp then. Good luck, let me know what u find Curt