Having the fan blow out will mostly repeatedly replace the air close by the window, as the air sucked by the fan is taken from the immediate surroundings, and that air is standing mostly still, so the same pocket of air will continuously be replaced.
If you let the fan blow into the room however, it will be pushed further into the room, helping to move the air in the room more, and increasing how much air is actually shifted out of the room.
The problem with taking in the air it blows from the immediate vicinity still exist, but normally there will be some movement in outside air, so the air sucked from the immediate vicinity is replaced by the natural wind replacing it outside, preventing the air that just escaped the room from being sucked in again.

So blowing in replaces way more air on average, the exception being if it works against an airflow in the room in opposite direction.

The air you’re pushing out needs to be replaced somehow. If there’s only one window open, then the only place for air to come in is through that same window, and it won’t work too well. But, if you have two windows open on opposite sides of the room, then that would work very well. If there is a second window open anywhere in the house, the air will be pulled through the house from that window.

But, also simply stirring it around will dilute it, and make it less noticeable.

Just fart out the window.

It will depend upon how the air is flowing outside. So if the wind is into the window then setting the fan to blow towards the fan will do nothing. .

To put this a different way

Imagine one side your house is on the north, and the other side house on the south. Now imagine you have windows on both north and south open. Now imagine the wind is coming from the south. So it is blowing into the south window. if you fart in that room and you point the fan towards the South window nothing is going to happen.

But if you turn the fan around and face it away from that window now you are using the air current in your favor.

Most rooms don’t have windows on both north and south sides. So open the doors to the appropriate rooms and open windows in both rooms. The wind will blow through the house. The smell will clear faster with a fan if your point it with the air current.

It’s all just basic physics

Fart directly into an air purifier and it’ll get removed even faster. Only works with the ones that use a carbon filter, I think.

It depends on if you set the fart on fire or just let it go.

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