This is pretty standard. Fan wires generally are only used when you want to run the blower without heat or ac. When heat is called for, the logic on the control board handles blower operation, as the precise order/timing will vary from furnace to furnace. On mine, for instance, when the thermostat calls for heat:

The small draw/exhaust blower runs for ~30sec to clear the combustion chamber

The ignitor heats

The gas valve opens and combustion starts

Wait~20-30 seconds

Then the main blower starts.

If your furnace is short cycling (shutting off before reaching the demand temp, despite the thermostat still calling for heat) this issue is likely at the furnace. Have you replaced your filter recently? Too low of airflow over the heat exchanger can cause a safety shutdown. Another common cause of short cycling is a flame sensor getting gunked up.