Flame goes out when fan kicks on

I have the same problem. The vent flu. Is building up condensation and running back into the furnace and cloggs the little black vent hose with water. Then it comes back on then off etc.
somehow the trick is to make as straight of an upward vent flu as possible ( definitely not 45 degree elbows and really not 60's either) seal all connections with glue.

That's usually your flame sensor. If the board doesn't detect a flame after burner is running it will shut your gas valve down. You can usually fix this by cleaning the flame sensor with some extra fine steel wool.

If your furnace has an ehxaust fan, your furnace does not have a thermocouple, it has a flame sensor. The flame sensor is a small metal rod that is in the burner area. It has one wire leading to it. Your problem seems to be flame sensor. If unit does not sense flame, it will kick off the gas valve.
Turn power off, take sensor out, (should be on 1/4" hex screw).. Probably some white film on sensor. Clean with steel wool, or dollar bill if don't have steel wool. Replace sensor.

The furnace is not sensing the flame. Turn off the power.
There is a metal rod in front of the last burner to light. It is corroded and needs to be cleaned. Its the flame sensor .
take the 1 screw out holding it in to the bar and scrub it with some steel wool till shiny. replace and turn on furnace. all should be fine.

On a call for heat, the 24 volt thermostat sends a signal to the control module. The control module will indicate a call for heat with a light on the control either blinking or remain solid depending upon model. The inducer (exhaust) blower will purge all gasses from the furnace and pressurize a pressure switch. Once the pressure switch tells the module to continue, the electronic ignition will energize and send 120 volts to the igniter. The igniter will glow and you will be able to see it if viewed thru the small inspection port. Once the igniter gets hot enough, it sends a signal to the module opening up the gas valve (24 volts). Either a pilot will come on or the burner tube will ignite then spread the flame to all burners. Lastly a safety sensor will be looking for a certain temperature within a few seconds and the furnace will continue to operate and the room air blower will turn on in a minute or two.

What could go wrong? The unit will not run if there is no signal from the thermostat (bad thermostat or broken wire), the control module does not sense a signal from the thermostat (bad control), the inducer does not energize (bad motor), the pressure switch does not close (blocked vent piping, bad switch, plugged condensate hose), the igniter does not energize (bad control, bad igniter), the gas valve does not open or there is no gas (bad gas valve, broken wire, no gas), the pilot does not light (dirty pilot), the burner does not light (bad burner, plugged orifice, not enough combustion air), the flame does not spread to each burner (bad flame spreader, dirty flame spreader, more bad burners), the flame safety sensor does not detect flame (dirty or bad flame spreader, bad flame sensor, broken wire, bad control), or the room air blower does not energize (bad fan motor, bad control).

On a call for heat, the 24 volt thermostat sends a signal to the control module. The control module will indicate a call for heat with a light on the control either blinking or remain solid depending upon model. The inducer (exhaust) blower will purge all gasses from the furnace and pressurize a pressure switch. Once the pressure switch tells the module to continue, the electronic ignition will energize and send 120 volts to the igniter. The igniter will glow and you will be able to see it if viewed thru the small inspection port. Once the igniter gets hot enough, it sends a signal to the module opening up the gas valve (24 volts). Either a pilot will come on or the burner tube will ignite then spread the flame to all burners. Lastly a safety sensor will be looking for a certain temperature within a few seconds and the furnace will continue to operate and the room air blower will turn on in a minute or two.

What could go wrong? The unit will not run if there is no signal from the thermostat (bad thermostat or broken wire), the control module does not sense a signal from the thermostat (bad control), the inducer does not energize (bad motor), the pressure switch does not close (blocked vent piping, bad switch, plugged condensate hose), the igniter does not energize (bad control, bad igniter), the gas valve does not open or there is no gas (bad gas valve, broken wire, no gas), the pilot does not light (dirty pilot), the burner does not light (bad burner, plugged orifice, not enough combustion air), the flame does not spread to each burner (bad flame spreader, dirty flame spreader, more bad burners), the flame safety sensor does not detect flame (dirty or bad flame spreader, bad flame sensor, broken wire, bad control), or the room air blower does not energize (bad fan motor, bad control).

If the burners do not fully ignite, the sensor will shut the furnace down. The lack of sufficient burner flame is a gas valve or gas pressure problem within the valve.
Replacing the gas valve with the original one may get you going. I surely suggest a professional service company for safety reasons and a thorough inspection. A 20 year old furnace is past it's normal life span.

watch your flame on the burners and see if the flame changes when the blower starts if it does and makes the flame jump around you have a busted heat exchanger

There are 4 -steps for full run time to kick in a furnace.. step-1, inducer fan runs & prove thru the air tube which closes a switch and tell the circuit board to go to step-2, the igniter lights the pilot & prove its lit by heating the thermocouple, which tells the circuit board to go to step-3, opens the gas valve to dump gas in, which will light from the pilot light, and prove by a flame sensor metal stick on the last burner , which heats up and tell the circuit board Saying, ok , I've got flame.. step-4. when the box heats up, it closes a switch and turn on the main fan...! (Some gas-valves has a brain -or circuit board on top of them and are real sensetive to moisture..)

yes! The furnace goes through 4--steps for complete run time. Step -1 the small vent fan comes on, and prove it thru the small rubber tube that sucks the switch close and tells the c.b. its turnning, Step 2.. the glow plug glow and 08 seconds later, Step -3 then the gas valve open and dump in gas ,which ignites and heat up the box. and proves it got flame through a flame sensor (pin) sticking in the flame .& Step -4, the box heats up to a predetermined temp. and turn on the fan! thats all. Thank-you-very-much!