Ispanağı ayıklarken kız kardeşim brokoliyi küçük parçalara böldü.

It has two main parts:

• Ispanağı ayıklarken = while sorting/picking over the spinach
• kız kardeşim brokoliyi küçük parçalara böldü = my sister divided the broccoli into small pieces

So the overall pattern is:

[while doing X] + [subject + object + verb]

This is a very common Turkish sentence shape.

Ayıklarken comes from the verb ayıklamak, which means things like:

• to sort
• to pick over
• to clean by removing unwanted parts

So with vegetables, it often means removing bad leaves, dirt, thick stems, and so on.

Grammar-wise, ayıklarken means while sorting / while picking over.

It is formed from:

• ayıkla- = verb stem
• -r
• -ken = while / when

So historically it is like ayıklar + ken, which becomes ayıklarken.

By itself, -ken does not show person. So ayıklarken does not tell you whether it means:

• while I was sorting
• while she was sorting
• while someone else was sorting

You understand that from context.

If Turkish wants to make it clearer, it can add a subject:

• Ben ıspanağı ayıklarken... = While I was sorting the spinach...
• Kız kardeşim ıspanağı ayıklarken... = While my sister was sorting the spinach...

So this form is naturally a little dependent on context.

Because ıspanağı is the definite direct object form of ıspanak.

In Turkish, a specific direct object often takes the accusative ending. Here, the spinach is a specific thing being sorted, so Turkish uses the accusative:

• ıspanak = spinach
• ıspanağı = the spinach / spinach as a specific direct object

Also, notice the sound change:

• ıspanak
• ıspanağı

The final k softens to ğ before a vowel-initial ending. That is very common in Turkish.

For the same basic reason: brokoliyi is also a definite direct object.

• brokoli = broccoli
• brokoliyi = the broccoli / broccoli as a specific direct object

The -i is the accusative ending here, and the y is just a buffer consonant. Turkish often inserts y when a vowel-initial suffix is added to a word that already ends in a vowel.

So:

• brokoli + i would be awkward
• Turkish makes it brokoliyi

Because of the ending -im.

• kız kardeş = sister
• kız kardeşim = my sister

That -im is the first person singular possessive suffix.

Turkish often does not need the separate word benim for my unless you want emphasis or extra clarity.

So:

• kız kardeşim = my sister
• benim kız kardeşim = my sister, but with extra emphasis or explicitness

Küçük parçalara means into small pieces.

Breakdown:

• küçük = small
• parça = piece
• parçalar = pieces
• parçalara = to/into pieces

Here, parçalara is in the dative form, and with bölmek this is very natural. Turkish often uses the dative to show the resulting form or what something is divided into.

So:

• brokoliyi = the thing being divided
• küçük parçalara = what it is being divided into

That means brokoliyi is the direct object, but küçük parçalara is not a second direct object. It is a complement showing the result.

Because Turkish normally puts the finite verb at or near the end of the sentence.

So this is very typical:

• time/background information first
• other sentence elements next
• main verb last

Here:

• Ispanağı ayıklarken = background/time
• kız kardeşim = subject
• brokoliyi = object
• küçük parçalara = result complement
• böldü = verb

This end-of-sentence verb position is one of the most important word-order habits in Turkish.

Yes. Turkish word order is more flexible than English, because the case endings show what each noun is doing.

For example, you could also say:

• Kız kardeşim ıspanağı ayıklarken brokoliyi küçük parçalara böldü.

That still makes sense.

However, the original version sounds very natural because it starts with the while... part and sets the scene first. Turkish often does that with time expressions and background actions.