Learn Turkish: Küçük çocuklar yetişkin gözetiminde olmak zorunda. - Small children have to be under adult supervision.

To express obligation using zorunda, you must put the main action in its infinitive form (-mAk) right before it. Because the action in this sentence is a state of being ("to be under supervision"), the required infinitive verb is olmak (to be).

Since the subject is plural (çocuklar), why isn't the ending zorundalar?

In Turkish, when a third-person plural subject is explicitly stated and already carries the plural suffix -lar (like çocuklar), it is very common to drop the plural suffix on the predicate. So zorunda is perfectly natural and sounds less repetitive here, though zorundalar would also be grammatically correct.

How exactly does the word gözetiminde break down?

It starts with the noun gözetim (supervision). Together with yetişkin (adult), it forms a noun compound: yetişkin gözetimi (adult supervision), where the -i is the possessive suffix linking the two concepts. To say "under" (literally "in") that supervision, we add the locative case -de. The -n- is just a buffer letter used before adding a spatial case to a possessive suffix. So: gözetim-i-n-de.

Could I say yetişkinlerin gözetiminde instead of yetişkin gözetiminde?