Şifre çarkının sıkışmasına rağmen kodu bulduk.

It has two main parts:

• Şifre çarkının sıkışmasına rağmen = despite the cipher wheel getting stuck
• kodu bulduk = we found the code

A very literal breakdown is:

• şifre çarkı = cipher wheel / code wheel
• şifre çarkının sıkışması = the cipher wheel’s getting stuck
• şifre çarkının sıkışmasına rağmen = despite the cipher wheel’s getting stuck
• kodu bulduk = we found the code

So the sentence is built as:

[despite X] + [main action]

This is a very common Turkish pattern.

Because çarkının is in the genitive form, roughly like English of the wheel or the wheel’s.

Here is the progression:

• çark = wheel
• şifre çarkı = cipher wheel / code wheel
  This is a noun compound, so the second noun takes the compound ending -ı.
• şifre çarkının = of the cipher wheel / the cipher wheel’s

In this sentence, şifre çarkının is connected to the verbal noun sıkışması:

• şifre çarkının sıkışması = the cipher wheel’s getting stuck

So çarkının is needed because Turkish often marks the subject of a verbal noun clause with the genitive.

sıkışmasına comes from the verb sıkışmak, which means to get stuck, to jam, or to become trapped depending on context.

It breaks down like this:

• sıkış- = get stuck / jam
• -ma = verbal noun ending, turning the verb into something like getting stuck
• -sı = a possessive-type ending used here because the verbal noun has a subject
• -na = dative ending to / toward

So:

• sıkışması = its getting stuck
• sıkışmasına = to its getting stuck

That may sound odd in English, but the dative is there because rağmen requires it.

Because rağmen is a postposition that means despite or in spite of, and it normally requires the word before it to be in the dative case.

So the pattern is:

• X-e rağmen = despite X

Examples:

• yağmura rağmen = despite the rain
• zorluğa rağmen = despite the difficulty
• sıkışmasına rağmen = despite its getting stuck

So in your sentence:

• şifre çarkının sıkışmasına rağmen = despite the cipher wheel getting stuck

This is a fixed and very common pattern in Turkish.

This is a very common thing in Turkish verbal noun clauses.

When Turkish turns a verb into a noun-like form such as getting stuck, it often marks who performs that action with:

• a genitive noun or pronoun for the subject
• a possessive ending on the verbal noun

So:

• şifre çarkının sıkışması
  literally: the cipher wheel’s getting stuck

This is not possession in the everyday sense. It is just how Turkish grammar connects the subject to a verbal noun.

You can think of it like this:

• çarkının tells you whose action
• sıkışması is the action treated as a noun

Together they mean the cipher wheel getting stuck.

Because kodu has the accusative ending, which usually marks a specific or definite direct object.

• kod = code
• kodu = the code / the specific code

So:

• kod bulduk = we found a code / we found code
• kodu bulduk = we found the code

In this sentence, the code is understood as a particular code, so kodu is the natural choice.

bulduk means we found.

It breaks down as:

• bul- = find
• -du- = past tense
• -k = we

So:

• buldum = I found
• buldun = you found
• buldu = he/she/it found
• bulduk = we found

This is straightforward simple past tense.

That is very normal in Turkish.

Turkish often puts subordinate or background information first, and the main action comes later, with the verb at the end.

So:

• Şifre çarkının sıkışmasına rağmen kodu bulduk.

This feels natural and neutral in Turkish.

English can also do something similar:

• Despite the cipher wheel getting stuck, we found the code.

Turkish word order is flexible, but the most neutral order usually puts the main verb last.

Yes. Turkish has several ways to express although / even though / despite.

Some close alternatives are:

• Şifre çarkı sıkışmasına rağmen kodu bulduk.
  Less formal-sounding structure, but the original with çarkının sıkışmasına is more fully grammatical and explicit.

• Şifre çarkı sıkıştığı hâlde kodu bulduk.
  = Although the cipher wheel got stuck, we found the code.

• Şifre çarkı sıkıştı ama kodu bulduk.
  = The cipher wheel got stuck, but we found the code.

• Şifre çarkı sıkışsa da kodu bulduk.
  = Even though the cipher wheel got stuck, we found the code.

Your original sentence is a bit more formal and compact because of -masına rağmen.

şifre can mean several related things in Turkish:

• password
• code
• cipher
• sometimes combination

The exact meaning depends on context.

So şifre çarkı could be understood as:

• cipher wheel
• code wheel
• possibly something like a combination dial/wheel, depending on the situation

In this sentence, the exact English choice depends on what object is being talked about, but grammatically the Turkish stays the same.

A very literal gloss would be:

• Şifre çarkının = the cipher wheel’s
• sıkışmasına = to its getting stuck
• rağmen = despite
• kodu = the code
• bulduk = we found

So the full literal sense is:

Despite the cipher wheel’s getting stuck, we found the code.

That sounds slightly unnatural in English, but it matches the Turkish grammar quite closely.