The exhaust blower starts for about 15 seconds then shuts off. several minutes later, the same thing happens - over and over

Goodman furnance the exhaust fan starts, igniter glows hot, all burners lite up & will burn for about a minute and shuts down . I think it senses the room blower does not start so it shuts down.

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).
I suspect you have a bad igniter, you did not mention if it glows or not or a blocked air intake or exhaust pipe. Also check the condensate hoses for a blockage. While you are looking, change the air filter.