I can say easily that head is a synonym for supervisor/leader because the Head of Admissions at a university is the leader in the group of people considering admissions.
Supervise (v) doesn't quite remain synonymous with control, direct, or lead. Although a thesaurus might say it is a synonym, it could also just mean that those two words are similar.
Think of all these words on a scale, with different meanings. If 'supervise' is the word you want, all of these fall under it, some warmer and some colder.
Control, for example. If you have a toy robot or car, you are controlling it with the remote. The thing being controlled has no free will whatsoever.
Direct is also not quite the same. Think of this as someone directing a play or movie. It may be a little closer to supervise than control since a director may advise the actors on how to improve technique.
Lead is a little farther away than both of the other words. Think of someone leading an expedition into the Sahara Desert. This person is implying that the others must follow his lead.
Overall, supervise tends to imply a non-interfering caretaker. If someone has to supervise a classroom of high schoolers, it is implied that they know the basics, that they shouldn't be destroying everything and starting fires. But the supervisor is needed in case something small happens, or if one of the kids has a question.