I don't know if anyone else has noticed this but in most buildings and most rooms, radiators are predominantly placed under a window. 

Now, in my eyes, that is the worst place to put them; hot air rises, reaches the window (which no matter how well insulated it's still letting out heat, in loose terms) and the thermal energy of the air disperses around the window area, thus not doing much to warm up the room. 

Am I wrong to think this? I mean, I can hold my hand close to my window and feel that it is colder there than at the other end of my room, but then again, my room does warm up when the radiator is on.

The reason is because the heat loss occurs mostly in the windows and the fenestration. The idea is that you would like the incoming air to be heated up. Also, it creates an air curtain that prevents more heat from being lost through these exposed areas. Finally, it makes the temperature of the room more or less uniform. If the heaters were placed at the center of the room, you would create a large temperature gradient, resulting in drafts and discomfort for the occupant.