I live in an old apartment where 2 electrical circuits have been installed, as was standard back then when people only had a few bulbs and maybe a radio set.
And the wiring isn't really built for modern loads either, nor were the legacy devices UL-tested in a world of widely used high power appliances. In my also-old cottage also with 2 circuits, the kitchen and bathroom circuits have been upgraded, but the rest of the house maybe not so much. We use a space heater on the kitchen circuit and just shut it off while cooking. Note that the layout is such that an extension cord is not needed.
The extension cord also suggests a problem with NEC Article 400 which forbids use of extension cords as a substitute for building wiring.
Our bathroom has no heater, but we leave the door wide open when not in use, and convection takes care of the rest. If you are habituated to keep the door closed, question that habit.
What would cause such a problem in the plug of an extension designed to pass 1800W when used with a medium, intermittent load with no large inrush current?
Could be a NEC 110.14(D) "torque terminals to spec with an actual torque screwdriver" problem if the socket was not torqued to spec and the real heat source is the screw terminal.
Could also be cherry-picked selective application of NEC 110.3(B) "Follow instructions and labeling". I have skepticism that the heater instructions say "OK to use with extension cords" and "OK to run unattended". I suspect it says the opposite, in boldface.
Cadet makes 750W baseboard heaters which hardwire in, which would eliminate the extension cord trouble, and are rated for unattended use. This is the way to go.