Do you have the jumpers between the RC and RH as shown in the 3-wire heating diagram?
When the fan switch is in auto, the furnace or a/c generally control the fan, an only indirectly from the thermostat. On heating, it turns on when the air by the heat exchanger gets warm (on many, but not all). On a/c, it often turns the fan on immediately when it calls for cooling. So, if you tried to fool the furnace into thinking it has an a/c unit attached, and you wanted the thermostat to control the fan being on/off, instead of the G terminal, move it to the Y terminal on both ends, and make sure that the thermostat is in the cooling mode. But, if it is wired per the 3-wire heating diagram (with the jumper), putting the fan switch to ON should turn the fan on ANY time the furnace is powered up. The G wire is an override from the furnace's normal control of the blower. When in AUTO mode, the lead is not used, the blower control is all within the furnace based on the thermostat calling for heat/cold, and other internal conditions.