I'm currently 31 and I sure had suppositories in my childhood.
Most of them were paracetamol suppositories that were very often prescribed to children running a fever. Those were mostly prescribed to young children, and tended to be less used amongst teenagers, most doctors and parents assuming teenagers were old enough to take it orally. I got them until I was 5 years old or so, but my parents quickly gave up because I fussed a lot everytime I had to receive one.
As a teenager, I was given some eucayptus based suppositories when I had a cold, not that often though. As I mentionned in another post, I realized that other classmates also got them, but I guess they were not prescribed on a big scale. I guess a lot of teenagers just refused, but unlike to my childhood years I acted much more like a well behaved young man in my teen years, so I accepted them, also because I got them pretty rarely.
When I was a child, I remember that my then late teenage cousin got one. She was 18, I was staying at my uncle's house for one week and she did a one day trip to Germany while I was there. She came back home late in the evening - I guess it was a school trip or so - with a bad cold. At some point, I asked something to my aunt and she said "wait a sec', honey. I have to check if we have a suppository left for your cousin". She left the living room and went rummaging in the bathroom's pharmacy shelve. I hated suppositories as a child and I hated when adults mentionned them so I just stayed in the living room, not even trying to see what would happen to her.