Sonunda oyunu bitirmeyi becerdik.

Here sonunda means finally, in the end, or at last.

It comes from son (end) plus endings that originally make it look like at its end, but in everyday Turkish sonunda is very commonly used as a fixed adverb meaning finally / in the end.

So in this sentence, it sets the scene: after some difficulty or delay, the action finally happened.

Because oyunu is the definite direct object form.

• oyun = game
• oyunu = the game (as a specific object being acted on)

The verb bitirmek means to finish something, so the thing being finished is the direct object. In Turkish, a specific/definite direct object usually takes the accusative ending.

So:

• oyun bitirmek = to finish a game / game-finishing in a general sense
• oyunu bitirmek = to finish the game / a specific game

This is a very common Turkish distinction:

• bitmek = to end, to be over, to get finished
  
  • intransitive: the thing ends by itself
• bitirmek = to finish (something)
  
  • transitive: someone finishes something

So:

• Oyun bitti. = The game ended / was over.
• Oyunu bitirdik. = We finished the game.

In your sentence, we are doing the finishing, so Turkish uses bitirmek, not bitmek.

Bitirmeyi is the verb bitirmek turned into a verbal noun, and then put into the accusative.

Breakdown:

• bitir- = finish
• -me = verbal noun marker
• -y- = buffer consonant
• -i = accusative ending

So bitirmeyi means something like:

• finishing
• to finish (as a noun-like action)

In this sentence, oyunu bitirmeyi means finishing the game / to finish the game, and that whole chunk is the object of becerdik.

Because after becermek, Turkish normally uses the -meyi / -mayı form.

Pattern:

• bir şeyi yapmayı becermek = to manage to do something

So:

• oyunu bitirmeyi becerdik = we managed to finish the game

If you used bitirmek instead, it would not fit this structure naturally.

This is one of those patterns you simply learn with the verb:

• yapmayı becermek
• anlatmayı becermek
• bulmayı becermek
• bitirmeyi becermek

That y is a buffer consonant.

Turkish often inserts y when a suffix beginning with a vowel is added after a form that already ends in a vowel.

Here:

• bitirme
  • -i would create a vowel clash
• so Turkish inserts y
• result: bitirmeyi

This is very common:

• yapma
  • -ı → yapmayı
• gelme
  • -i → gelmeyi
• bitirme
  • -i → bitirmeyi

Becerdik comes from becermek, which means to manage, to succeed in doing, or sometimes to pull off.

So oyunu bitirmeyi becerdik means:

• We managed to finish the game
• We succeeded in finishing the game

There is often a nuance of:

• difficulty
• effort
• maybe even a little surprise that it was possible

Compared with a plain we finished the game, becerdik suggests it was not easy.

It breaks down like this:

• becer- = manage / succeed
• -di = past tense
• -k = we

So:

• becerdim = I managed
• becerdin = you managed
• becerdi = he/she/it managed
• becerdik = we managed

In this sentence, it is simple past: we managed.

Because they are doing two different jobs, and both happen to be in the accusative.

1. oyunu
   This is the object of bitirmek:
   finish what? → the game

2. bitirmeyi
   This whole verbal noun phrase is the object of becermek:
   manage what? → finishing the game

So the structure is roughly:

• [oyunu bitirmeyi] becerdik
• We managed [finishing the game]

In more natural English, we say We managed to finish the game, but Turkish builds it differently.

The most neutral order here is:

• Sonunda = finally
• oyunu = the game
• bitirmeyi = finishing / to finish
• becerdik = we managed

Turkish often puts the main verb at the end, so becerdik comes last.

A very literal gloss would be:

• Finally, the game, finishing-it, we-managed.

That sounds strange in English, but it is normal Turkish structure.

Yes. Turkish word order is flexible, although some orders sound more neutral than others.

The sentence you have is very natural. But Turkish can move parts around for emphasis:

• Oyunu sonunda bitirmeyi becerdik.
• Bitirmeyi sonunda becerdik.
• Sonunda bitirmeyi becerdik.
  (if the game is already understood from context)

The version Sonunda oyunu bitirmeyi becerdik is a good neutral sentence with emphasis on finally.

Both can often be translated as to manage or to succeed, but there is a slight difference in feel.

• başarmak = more neutral, often to succeed / achieve
• becermek = often to manage to do, sometimes with a stronger sense of difficulty, skill, or even somehow pulling it off

So:

• Oyunu bitirmeyi başardık. = We succeeded in finishing the game.
• Oyunu bitirmeyi becerdik. = We managed to finish the game.

In many situations they are close in meaning, but becermek can sound a bit more everyday and can suggest it wasn’t easy.

Oyunu bitirdik simply means:

• We finished the game.

It states the fact.

Oyunu bitirmeyi becerdik means:

• We managed to finish the game.

This adds extra meaning:

• it was difficult,
• it took effort,
• maybe it was uncertain whether it could be done.

So becermek adds a sense of achievement that plain bitirdik does not necessarily have.

Yes. Oyunu bitirebildik means We were able to finish the game.

This is close in meaning, but the nuance is a little different:

• bitirebildik focuses on ability/possibility
• bitirmeyi becerdik focuses more on managing to do it successfully

In many contexts, both would work.
But becerdik can feel a little more like we pulled it off.

Yes, it is natural and idiomatic.

Sonunda oyunu bitirmeyi becerdik sounds like something a native speaker would say after struggling with a game, level, task, or challenge.

It gives a very natural sense of:

• After all that, we finally managed to finish the game.