Learn Turkish: Uyuşturucu kullanmak çok tehlikeli.

No, uyuşturucu refers strictly to illegal drugs or narcotics. It literally translates to something that numbs, coming from the verb uyuşturmak (to numb or anesthetize). For medical drugs or prescriptions, you must use the word ilaç.

Why is the full infinitive kullanmak used here rather than the shorter kullanma verbal noun?

When an action acts as the general subject of a sentence, as Using drugs does here, Turkish uses the full -mAk infinitive. You generally only switch to the shorter -mA verbal noun when you need to attach possessive endings or case suffixes.

How would we apply this lesson's grammar to say In case of using drugs...?

You would drop the -mAk infinitive and use the -mA verbal noun. Because this is a general statement without a specific subject, it is most natural to use the passive: Uyuşturucu kullanılması halinde... (In the event of drugs being used...). You attach the third-person possessive -sI to the passive verbal noun before adding halinde or durumunda.