Oyun yazarı provayı izliyor.

Because Turkish often makes noun compounds by putting two nouns together.

• oyun = game, play
• yazar = writer
• oyun yazarı = playwright, literally play writer

In this kind of compound, the second noun usually takes a possessive-type ending. That is why it is yazarı, not just yazar.

So:

• oyun yazarı = playwright
• similar pattern: Türk kahvesi = Turkish coffee, okul kapısı = school door

It is actually built from yazar + compound ending -(s)I.

Since yazar ends in a consonant, the ending appears as -ı here:

• yazar
  • -ı → yazarı

This is the normal form in a noun compound. It does not mean his/her writer in this sentence; it is just part of the compound oyun yazarı.

Because prova is the direct object of the verb, and here it is a specific one, so Turkish uses the accusative ending.

• base form: prova
• accusative: provayı

The accusative ending here is -yı:

• the vowel matches vowel harmony
• the y is a buffer letter, added because prova already ends in a vowel

So:

• prova = a rehearsal / rehearsal in a general sense
• provayı = the rehearsal

Turkish usually marks a specific/definite direct object with the accusative.

So compare:

• Prova izliyor. = He/she is watching rehearsal / a rehearsal.
• Provayı izliyor. = He/she is watching the rehearsal.

In your sentence, the rehearsal is treated as specific, so prova takes the accusative ending.

İzliyor is the present continuous form, often used for actions happening now.

It comes from:

• izle- = to watch
• -iyor = present continuous ending

So:

• izliyor = he/she/it is watching

This form can also sometimes be used for near-future or current ongoing situations, depending on context.

Turkish often leaves subject pronouns out because the verb ending already gives enough information.

• izliyorum = I am watching
• izliyorsun = you are watching
• izliyor = he/she/it is watching
• izliyoruz = we are watching

Since the subject is already stated as oyun yazarı, there is no need to add o.

Turkish default word order is often Subject + Object + Verb.

So here:

• Oyun yazarı = subject
• provayı = object
• izliyor = verb

That gives:

• Oyun yazarı provayı izliyor.

This is very normal Turkish word order.
Turkish can move things around for emphasis, but the verb often stays at the end.

Yes, Turkish word order is flexible, but different orders change emphasis.

For example:

• Oyun yazarı provayı izliyor. = neutral/basic order
• Provayı oyun yazarı izliyor. = emphasizes the playwright
• İzliyor oyun yazarı provayı. = possible in special contexts, but less neutral

A learner should usually start with the standard Subject + Object + Verb order.

It can mean both, depending on context.

• oyun = game
• oyun = play (theatrical play)

In oyun yazarı, it clearly means play, so the whole phrase means playwright.

Context is very important with this word.

Not usually if you specifically mean playwright.

The standard expression is:

• oyun yazarı = playwright

Tiyatro yazarı sounds less standard and may be interpreted differently. If you want the normal Turkish term for someone who writes plays, oyun yazarı is the best choice.

The verb stem is izle-, but when Turkish adds the present continuous ending -iyor, the form becomes:

• izle- + -iyor → izliyor

This kind of vowel change is normal in Turkish.

Pronunciation is roughly:

• izliyor ≈ iz-lee-yor

Not izleyor.

No. Turkish does not have separate words exactly like English a and the.

Instead, definiteness is often shown in other ways, especially with the object:

• prova = a rehearsal / rehearsal
• provayı = the rehearsal

So the accusative ending on provayı helps give the sense of the here.

Here is the breakdown:

• oyun — noun
• yazarı — noun, part of a noun compound
• provayı — noun in the accusative case
• izliyor — verb, present continuous, third person singular

So the sentence structure is:

• [subject noun phrase] [object noun phrase] [verb]