Learn Turkish: Doktor yanağını göstermesini istedi. - The doctor asked him to show his cheek.

In Turkish, to say you want someone else to do something, you use a verbal noun (the -me or -ma form) with a possessive ending, followed by the accusative case. Literally, the doctor wanted 'his showing'. So gösterme (showing) gets the possessive -si (his) and then the accusative -ni because it is the direct object of istedi (wanted). We translate it as 'asked' here because a doctor wanting a patient to do something is naturally understood as a request.

What is the root word of yanağını and why does it have a ğ?

The root word is yanak (cheek). When a suffix starting with a vowel is added to a noun ending in k, that k softens to a ğ (soft g). Since the possessive suffix -ı is added to mean 'his cheek', yanak becomes yanağı.

Both yanağını and göstermesini have an n before their final vowel. Why is that?

That is the buffer letter n. In Turkish, whenever you add a case ending (like the accusative -ı or -i) to a third-person possessive suffix (like -ı in yanağı or -si in göstermesi), you must insert an n between them. So we get yanağı-n-ı and göstermesi-n-i.