Turnikede kartım çalışmadığı için görevli beni başka kapıdan geçirdi.

A useful breakdown is:

• Turnikede = at/in the turnstile
• kartım = my card
• çalışmadığı için = because it did not work
• görevli = the staff member / attendant
• beni = me
• başka kapıdan = through another gate/door
• geçirdi = made me pass / let me through

So the structure is basically:

Because my card didn’t work at the turnstile, the attendant let me through another gate.

Turnike means turnstile. The ending -de is the locative suffix, which often means in, at, on depending on context.

• turnike = turnstile
• turnikede = at the turnstile

Here it tells you where the problem happened: the card did not work at the turnstile.

Kartım is:

• kart = card
• -ım = my

So kartım means my card.

This is a possessive ending, very common in Turkish:

• kartım = my card
• kartın = your card
• kartı = his/her/its card

Because için usually attaches to a noun-like clause, not directly to a fully conjugated finite verb in this kind of structure.

Here is the logic:

• çalışmak = to work/function
• çalışma- = not work
• çalışmadığı = the fact that it didn’t work / its not working
• çalışmadığı için = because it didn’t work

This pattern is extremely common in Turkish:

• geldiği için = because he/she came
• hasta olduğu için = because he/she is sick
• vakit olmadığı için = because there is no time

So çalışmadığı için is a natural grammatical way to say because it didn’t work.

That -ı is part of the subordinate clause ending. In a form like çalışmadığı, Turkish is turning the verb into a modifier/noun-like structure.

Very roughly:

• çalışma = not working
• çalışmadığı = that it did not work / its not having worked

There is also an implied third-person possessor in this structure, because the thing that did not work is kartım. Turkish often uses this pattern in subordinate clauses.

You do not need to translate every tiny piece literally each time. It is best learned as a common pattern:

• Verb + -DIK + possessive + için
  = because ... did

So:

• çalışmadığı için = because it didn’t work

Turkish has no definite article like English the.

So görevli can mean:

• a staff member
• the staff member
• the attendant

Which one sounds best depends on context. In this sentence, English naturally prefers the attendant or a staff member.

Also, görevli literally means something like the person on duty / staff member / official.

Because beni is the accusative form of ben.

• ben = I
• beni = me

Here, beni is the direct object of geçirdi. The attendant did something to me: he/she let me through.

Compare:

• Ben geldim. = I came.
• Görevli beni çağırdı. = The attendant called me.

So beni is required here.

Başka kapıdan means through another gate/door.

Breakdown:

• başka = other / another
• kapı = door, gate
• kapıdan = from the door / through the door

The suffix -dan / -den is usually called the ablative suffix. Its basic meaning is from, but with movement verbs it can also correspond to English through, especially in this kind of context.

So:

• başka kapıdan geçmek = to pass through another gate

Even though the suffix literally looks like from, the natural English translation here is through another gate.

This is one of the most important points in the sentence.

• geçmek = to pass, go through
• geçirmek = to make someone pass, put someone through, let someone through

So:

• Ben başka kapıdan geçtim. = I went through another gate.
• Görevli beni başka kapıdan geçirdi. = The attendant let me through another gate.

The verb geçirdi is the causative form. It shows that the attendant caused or enabled the action.

In English, depending on context, you might translate it as:

• let me through
• sent me through
• made me go through

Here let me through sounds the most natural.

Turkish word order is often Subject–Object–Verb, and reasons often come before the main clause.

Here the structure is:

[Reason clause] + [subject] + [object] + [place phrase] + [verb]

So:

• Turnikede kartım çalışmadığı için = because my card didn’t work at the turnstile
• görevli = the attendant
• beni = me
• başka kapıdan = through another gate
• geçirdi = let through

The verb usually comes at the end, which is very normal in Turkish.

Yes, you could reorder parts somewhat, because Turkish word order is flexible.

For example:

• Turnikede kartım çalışmadığı için görevli beni başka kapıdan geçirdi.
• Kartım turnikede çalışmadığı için görevli beni başka kapıdan geçirdi.

Both are understandable. The difference is mainly focus:

• Turnikede kartım çalışmadığı için... puts early emphasis on where the problem happened.
• Kartım turnikede çalışmadığı için... starts with my card, so the card is slightly more prominent.

Both are natural, but the original sentence sounds very normal.

Yes. In Turkish, çalışmak can mean not only to work in the sense of having a job, but also to function / operate.

So for machines, devices, cards, systems, etc.:

• Makine çalışıyor. = The machine is working.
• İnternet çalışmıyor. = The internet isn’t working.
• Kartım çalışmadı. = My card didn’t work.

So this is a very normal use of çalışmak.