Damat alyansı unuttuğunu sanmış, ama alyans çantadaymış.

Turkish often repeats the noun where English might use it.

• alyansı unuttuğunu sanmış = he thought he had forgotten the ring
• ama alyans çantadaymış = but the ring was in the bag

You could say ama o çantadaymış, but repeating alyans is very natural and keeps the sentence clear.

Because the first alyansı is a definite direct object, so it takes the accusative ending -ı.

• alyans = ring
• alyansı = the ring as a specific object

In the second clause, alyans is the subject, not the object, so there is no accusative ending:

• Alyans çantadaymış = The ring was in the bag

unuttuğunu is the subordinate-clause form meaning something like that he forgot / that he had forgotten.

It is built from unutmak (to forget) using the very common Turkish -DIK clause pattern. In this sentence, the whole phrase

• alyansı unuttuğunu

means

• that he forgot the ring
• or in smoother English here, that he had forgotten the ring

So:

• Damat alyansı unuttuğunu sanmış = The groom thought that he had forgotten the ring

Because Turkish usually does not use a separate conjunction like English that in this kind of sentence. Instead, it turns the verb into a noun-like subordinate clause.

So English:

• He thought that he had forgotten the ring

becomes Turkish:

• Alyansı unuttuğunu sanmış

Literally, it works more like:

• He thought [his forgetting the ring]

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

Because in this kind of structure, if the embedded clause has no separately stated subject, it is usually understood to have the same subject as the main clause.

So here:

• main subject = damat
• understood subject of unuttuğunu = also damat

If it were someone else, Turkish would usually show that:

• Damat, gelinin alyansı unuttuğunu sanmış. = The groom thought that the bride had forgotten the ring.

Yes, sanmak means to think, to suppose, or to assume.

In many contexts, sanmak suggests that the person’s belief may be mistaken, which fits this sentence well, because the second clause corrects the idea:

• He thought he had forgotten the ring, but actually it was in the bag.

So sanmış is a very good choice here.

This is the evidential/inferential past in Turkish.

Very roughly, it can imply things like:

• apparently
• it turned out that
• I learned later that
• or just a storytelling/narrative past

So:

• sanmış = he apparently thought / he had thought
• çantadaymış = it turned out it was in the bag

In natural English, you often won’t translate -mış every time, but the nuance is there.

Also, -ymış appears after a noun or location expression like çantada. The y is just a buffer sound:

• çanta-da-y-mış

It breaks down like this:

• çanta = bag
• -da = in / at / on
• -ymış = was apparently / it turned out to be

So:

• çantadaymış = it was in the bag / it turned out to be in the bag

Turkish does not need a separate word for it was here.

Either is possible from the Turkish alone.

Turkish has no articles like a or the, so çantada can mean:

• in the bag
• in a bag

Context decides. In this sentence, English will often say in the bag because it sounds like a specific, relevant bag.

Yes:

• Damat alyansı unuttuğunu sandı, ama alyans çantadaydı.

This would also mean basically the same thing.

The difference is nuance:

• sandI / çantadaydı = more plain, direct past
• sanmış / çantadaymış = more narrative, inferential, or as it turned out

So the original sentence has a slight storytelling/discovery flavor: The groom thought he had forgotten the ring, but as it turned out, the ring was in the bag.

Because Turkish normally puts the finite verb at the end of the clause.

So in:

• Damat alyansı unuttuğunu sanmış

the full object/complement alyansı unuttuğunu comes before the final verb sanmış.

That is very typical Turkish word order:

• [subject] [object/complement] [verb]

Even when the object is a whole subordinate clause, it usually still comes before the main verb.