Air coming out of electrical outlets when bathroom exhaust fan is running?
Without seeing it, it's hard to say for sure. However, the fan is creating 'negative pressure' inside your apartment (which appears to be pretty close to air tight), and so that wants to equalise with the outside any way it can. Thus, air is coming into the rooms by the easiest route it can.

I'd imagine the wall your windy sockets are on is either a stud+plaster (drywall?), roughly hollow sort of thing, or else the sockets have a channel in the solid wall through to the sub-floor, or somewhere else. The air is coming from there to equalise the pressure in your apartment.

As for solutions, they probably depend on your local building regulations and the rules for changes in your block. It feels like you could remove the socket, and the back-box that holds it in the wall and put some insulation behind (or in the channel for the cables). That would probably slow the airflow, but be aware it'll start coming through some place else in your property instead.

You may be better off investigating if you can put an air vent somewhere (your windows may have some at the top which you sort of slide to open and close?), or to put an explicit hole, grilles and maybe baffles somewhere (perhaps in the bathroom outside wall? or maybe the front door into shared areas?). The idea being that it creates a deliberately 'weak' point in the air-tightness of your property to allow the fan to suck the air out and for that air to be replaced relatively easily. It would probably increase the effectiveness of the fan, at the expense of letting (potentially cold) outside air in, but at least the air comes from a known source, and hopefully more easily than through random weak spots in unexpected places.