Gravity grille not opening when bathroom fan is on
The outside grille should definitely open when the fan is running. However, it isn't clear whether this is a cause or an effect of your airflow problem:


The grille may be stuck, in which case you need to lubricate/free/unbind it
There may be some other issue with the airflow, in which case nothing's pushing the grille open


First stop: check the grill manually so that you can see if they're working (they should open easily). If not, you've found the problem. If so, then you'll need to check other components (the ducting, the fan, etc).