Learn Turkish: Pilotun kendi uçağını radarda görmesi lazım. - The pilot needs to see his own airplane on the radar.

In Turkish, to express that someone needs to do something using lazım, you literally say that their doing of the action is necessary. The action is turned into a verbal noun using -me (görme), and the person doing it becomes the owner of that action. Therefore, we use the genitive case for the subject (pilotun) and the possessive for the verb (görmesi), literally creating the structure the pilot's seeing is necessary.

Why does uçak change to uçağını here?

It takes two suffixes separated by a buffer letter. First, the -ı is the 3rd person possessive (his airplane), which causes the k in uçak to soften to ğ. Because his airplane is the specific direct object of the verbal noun görmesi, it needs the accusative case -ı. The -n- in the middle is the buffer letter required between a 3rd person possessive suffix and any case marker.

Could we just say uçağını without kendi?

Yes, uçağını alone already means his airplane. Adding kendi (own) right before it emphasizes that it is his own airplane rather than someone else's. At the B2 level, you will frequently see kendi used alongside possessive suffixes to add this emphasis or to clarify exactly whose object is being discussed.

Does the word order matter inside the embedded clause? Could it be ?