Now I am even more confused… why can’t you go north from the laundry room? (assuming up is “north” in your picture).
Looks like that image is the dryer itself, right? It is on an outside wall (even if the wall itself has no window). That wouldn’t just be a solution, it would be the preferred solution because you can keeping the ductwork as short as possible (less condensation as the air cools and dew forms on the inside of the vent.
The other option would be a “U” shaped vent which went into the garage as shown with the red dot, and then a 180 degree bend to go out the exterior wall of the garage. 180 degrees is not ideal, but looks like it could be done in a short distance.