Turnikeden geçmeden önce çantamdaki fotokopiyi memura gösterdim.

Because the suffix -den / -dan / -ten / -tan is the ablative case, and with geçmek Turkish often uses the ablative for the thing you pass through / across / by.

• turnike = turnstile
• turnikeden geçmek = to pass through the turnstile

So turnikeden means something like through/from the turnstile in Turkish structure, even though natural English says through the turnstile.

The form is:

• turnike + den → turnikeden

The d version is used because of Turkish sound rules.

This is a very common Turkish pattern:

• verb + -madan / -meden + önce = before doing ...

So:

• geçmek = to pass
• geçmeden önce = before passing / before going through

This can look confusing at first, because -madan / -meden by itself often means without doing:

• yemeden çıktım = I left without eating

But when you add önce, the whole expression means before doing:

• yemeden önce = before eating
• gitmeden önce = before going
• geçmeden önce = before passing through

It breaks down like this:

• geç- = verb stem, pass
• -me- = negative marker
• -den = an adverbial ending here, giving a form used in expressions like -meden önce

So literally it looks like without passing, but in the fixed pattern -meden önce, it means before passing.

This is one of those places where it is better to learn the whole chunk:

• -madan/-meden önce = before doing

rather than trying to translate each piece literally every time.

Çantamdaki means the one in my bag or that is in my bag. It is built step by step:

• çanta = bag
• çantam = my bag
• çantamda = in my bag
• çantamdaki = the one that is in my bag

The suffix -ki turns a location phrase into something that can describe a noun.

So:

• çantamdaki fotokopi = the photocopy in my bag
• literally: the in-my-bag photocopy

This is very common in Turkish:

• masadaki kitap = the book on the table
• odadaki insanlar = the people in the room

Because it is a specific direct object, so Turkish uses the accusative.

• fotokopi = a photocopy / photocopy in a general sense
• fotokopiyi = the photocopy, a specific photocopy

In this sentence, the speaker is talking about a particular photocopy: the one in their bag.

Breakdown:

• fotokopi + yi → fotokopiyi

So:

• çantamdaki fotokopiyi = the photocopy in my bag

If you said fotokopi gösterdim, that would sound more like I showed a photocopy / some photocopy, less specific.

Because göstermek usually works like this:

• birine bir şeyi göstermek = to show something to someone

So the person receiving the thing is marked with the dative case:

• memur = officer / clerk / official
• memura = to the officer

The pattern in the sentence is:

• fotokopiyi = the thing being shown
• memura = the person it is shown to

Gösterdim means I showed.

Breakdown:

• göster- = show
• -di- = past tense
• -m = I

So:

• gösterdim = I showed

This is the definite/simple past in Turkish, used for completed actions in the past.

Because Turkish often drops the subject pronoun when it is already clear from the verb ending.

Here:

• gösterdim already means I showed

So adding ben is usually unnecessary unless you want emphasis.

• Ben ... gösterdim = I showed ...
  This would sound more emphatic, like contrasting with someone else.

In normal Turkish, leaving out ben is very natural.

No, Turkish word order is flexible, but the given order is very natural.

The sentence is:

• Turnikeden geçmeden önce çantamdaki fotokopiyi memura gösterdim.

A rough structure is:

• time expression: Turnikeden geçmeden önce
• object: çantamdaki fotokopiyi
• indirect object: memura
• verb: gösterdim

Turkish often puts the verb at the end. Other orders are possible, for example:

• Turnikeden geçmeden önce memura çantamdaki fotokopiyi gösterdim.

That is also natural. Changing the order can slightly change what is emphasized, but the basic meaning stays the same.

Yes, that is the best natural translation.

More literally, it is something like:

• the photocopy that is in my bag

So:

• çantamdaki = in-my-bag / that is in my bag
• fotokopiyi = the photocopy

Together:

• çantamdaki fotokopiyi = the photocopy in my bag

Because Turkish often uses suffixes instead of separate relative words like that, which, or who.

In English, you might say:

• the photocopy that is in my bag

In Turkish, that relationship is packed into -ki:

• çantamdaki fotokopi

So instead of a separate mini-clause, Turkish often creates a compact adjective-like form attached directly to the noun.

That is why Turkish can look very dense at first: a lot of meaning is built into suffixes rather than separate words.