When I pour coolant into my 1991 Mazda Proteges radiator, it starts to fill then I see it leaking out of the front of motor. What could be the issue?

Where is your coolant leak coming from? Front of vehicle is possible radiator damage ; if from under or behind the motor this can be rubber engine hoses or heater core hose deterioration

It could be anywhere
But let it cool and fill with water see if you can see it leaking. If not fill it to top put cap back on and squeeze upper radiator hose to put pressure in system.
If you don't see anything it could be water pump

Take the radiator cap off and start the vehicle. When it gets hot enough to where the thermostat would open you should be able to see the coolant flowing if the pump is working.
Normally a raditor going bad would be leaking. It could get some rows plugged up if a person dumped stop leak in.

But if that has one of the 3 liter V6 those are grate for having an air pocket that will not burp out on its own. If the coolant was drained or low and got filled it could have an air pocket.

What you would need to do is remove the heater hose at the back of the motor and fill it with coolant there.Should also be parked on level ground or down hill so the back of the motor where you're filling it is at a high point.

Check the bottom side of the water pump for wetness, as well as the lower radiator hose. The water pump if leaking, will run the coolant down the front of the motor,and can appear to be coming from the top of the oil pan. It is nearly impossible for the top of the oil pan to leak coolant.

TO CHANGE RADIATOR FIRST YOU TURN OFF RADIO FIRST TO KEEP FROM LOSING RADIO CODE.DISCONNECT NEGATIVE BATTERY CABLE.RAISE FRONT OF VECHICLE SUPPORT ON JACK STANDS.REMOVE THE LOWER SPLASH SHIELDS.NOW YOU NEED TO DRAIN THE COOLING SYSTEM.CATCH COOLANT IN A LARGE DRAIN PAN.REUSE COOLANT IF ITS FAIRLY NEW.IF COOLANT VERY OLD CHANGE IT.PUT OLD COOLANT IN SOME EMPTY CONTAINER TAKE TO A GARAGE OR WHOEVER DISPOSE IT.DONT POUR ON GROUND.HARMFUL TO THE ENVIRONMENT.MOVE THE AIR INTAKE DUCT PASSES OVER THE TOP RADIATOR HOSE.NOW WHEN ALL COOLANT DRAIN OUT.LOOSEN UPPER AND LOWER RADIATOR HOSE CLAMPS.THEN DETACH THE RADIATOR HOSES FROM THE FITTINGS.IF THEY ARE STUCK GRASP EACH HOSE NEAR THE END WITH A PAIR OF ADJUSTABLE PLIERS AND TWIST IT TO BREAK THE SEAL THEN PULL IT OFF REMOVE SMALL COOLANT SURGE LINES FROM TOP OF RADIATOR THAT GOES TO THE OVER FLOW RESERVOIR..DISCONNECT COOLING FAN ELECTRICAL CONNECTOR.THEN REMOVE THE COOLING FAN.REMOVE DISCONNECT AUTOMATIC TRANSMISSION COOLER LINES GET READY TO CATCH TRANSMISSION FLUID THEN PLUG LINES. NOW UNSCREW THE FOUR BOLTS TWO ON EACH SIDE OF THE RADIATOR THAT ATTACH RADIATOR TO THE AIR CONDITIONER CONDENSOR CAREFULLY TIP TOP OF THE RADIATOR SLIGHTLY TOWARDS THE ENGINE.REMOVE THE UPPER RADIATOR SEAL AND LIFT RADIATOR FROM THE ENGINE.NOW IS THE TIME TO REPLACE THE UPPER AND LOWER RADIATOR HOSES WITH NEW ONES SINCE ALL COOLANT REMOVE I WOULD REPLACE THERMOSTAT AND RADIATOR PRESSURE CAP.NOW WHEN YOU GET READY TO INSTALL NEW RADIATOR USE SHEETS OF CARD BOARD PROTECT RADIATOR FINS AND CORES FROM DAMAGE WHILE SITTING RADIATOR BACK IN PLACE.ONCE YOU GET UPPER AND LOWER HOSES IN PLACE ALSO PLACE SMALL SURGE HOSES FROM THE OVER FLOW RESERVOIR TO RADIATOR NEW THERMOSTAT AND TRANSMISSION COOLER LINES ALL HOOK UP. START POURING ANTIFREEZE IN THE RADIATOR OR OVERFLOW RESERVOIR IF RADIATOR DONT HAVE RADIATOR CAP ON IT.POUR A FULL GALLON ANTIFREEZE AND A HALF MORE IF LONG AS THE COOLANT STAYS ON THE COLD LEVEL MARK ON OVERFLOW RESERVOIR DONT OVER FILL.IF YOU POUR ANTIFREEZE IN RADIATOR KEEP POURING ANTIFREEZE UNTIL LEVEL STOP DROPPING AND LEVEL OFF AT RADIATOR TOP.START ENGINE CHECK FOR LEAKS. LET ENGINE RUN UNTIL IT WARM UP TOP RADIATOR HOSE GET HOT.WATCH COOLANT LEVEL IN RADIATOR OR COOLANT OVERFLOW RESERVOIR WHEN THERMOSTAT OPENS COOLANT LEVEL WILL DROP.ONCE YOU HAVE ENOUGH COOLANT IN THE SYSTEM COOLANT LEVEL WILL STOP DROPPING.THEN YOU READY TO GO.LET ENGINE SIT A LITTLE WHILE CHECK COOLANT LEVEL AGAIN.COOLANT LEVEL IN THE OVER FLOW RESERVOIR SHOULD BE AT HOT MARK FOR HOT ENGINE AND WHEN ENGINE COLD COOLANT LEVEL SHOULD BE AT THE COLD MARK.IF POUR ANTIFREEZE IN THE RADIATOR COOLANT LEVEL SHOULD BE AT TOP OF RADIATOR BELOW PRESSURE CAP.

If water poured out you probably busted a hose or radiator sprung a leak. Sounds like that and a bad thermostat or water pump went out.

This is how I like to do it. Make sure the car is parked with the front end at an upward angle. Drain the coolant from the petcock at the bottom of the radiator. Remove the upper and lower hoses at the radiator. Remove the thermostat housing, and the heater core hoses at the fire wall. Flush out the radiator by pouring fresh water into the overflow, and the hole where the thermostat was. Flush the heater core as best you can by pouring fresh water into the heater core hose connections.
Let all the water drain out, then button everything back up and replace the thermostat with a new one. Refill the rad and engine by filling the coolant reservoir with a mix of 50% water and 50% antifreeze, or buy the premixed stuff from the auto parts store. (Buy 4 jugs of premix or 2 of straight antifreeze)
Run the engine until it reaches operating temperature and the upper rad hose gets hot. Rev the engine a little to help get the air out of the cooling system. When the thermostat opens, the coolant level will drop in the reservoir, so top it up and replace the cap.
Check for leaks, and check the coolant level for the next few days, and check for leaks.
Dispose of the old coolant responsibly

your electric cooling fan is not switching on that is either 1. you fan is bad which you can test jumping a hot wire to the pos side of the fan if it works then exclude it. 2. your fan works on a ground system there is a coolant fan switch in the head that can cause no fans, and the last 3. nfan relay or fuse bad

Heater core must be leaking. Turn it on defrost and see if the windshield fogs up.

Fill up the cooling system and start the jeep. Let the jeep get at operating temp with the radiator cap off. If the coolant level goes down, add more. Keep your eye on the radiator level and see if any bubbles appear while the car is running. DON'T keep your head over the filler neck, if you do see alot of bubbles or air bubbles, you could have a bad headgasket.