Cumartesi akşamı eniştemle yengemi ağırlarken yeğenim başucundaki oyuncağını göstermeden edemedi.

Because Turkish often uses bare time expressions adverbially.

• Cumartesi = Saturday
• akşamı = its evening / the evening of it

So Cumartesi akşamı means on Saturday evening or Saturday evening.

You do not need -de here. Turkish commonly says:

• Pazartesi sabahı = Monday morning
• dün akşam = yesterday evening
• gelecek hafta = next week

Using -de would usually change the meaning or sound unnatural in this kind of time expression.

Here -le is functioning like and, not really with in the English sense.

So:

• eniştemle yengemi = my brother-in-law and my sister-in-law

In Turkish, one noun can take -le/-la and the other noun can carry the case ending for the whole pair:

• Ahmet'le Ayşe'yi gördüm = I saw Ahmet and Ayşe
• annemle babamı aradım = I called my mother and father

So in your sentence, yengemi has the accusative ending, and eniştemle is linked to it as part of the same object phrase.

Because in coordinated phrases like this, Turkish often marks the case only once, usually on the last item.

So:

• eniştemle yengemi literally works like
• my brother-in-law and my sister-in-law as one combined object

The accusative -i on yengemi covers the whole pair.

This is normal and very common in Turkish.

ağırlarken comes from:

• ağırlamak = to host, entertain, receive as a guest
• -ken = while, when

So:

• ağırlarken = while hosting / while entertaining

It is a while-clause form. Similar examples:

• yemek yerken = while eating
• konuşurken = while speaking
• çalışırken = while working

So eniştemle yengemi ağırlarken means while hosting my brother-in-law and sister-in-law.

Not necessarily.

That is one thing English speakers often notice: -ken clauses usually do not show the subject directly unless the sentence makes it explicit.

Here the main clause subject is clear:

• yeğenim ... edemedi = my nephew/niece couldn't help...

But in ağırlarken, the subject is omitted. You understand it from context. Most naturally here, it means:

• while I was hosting my brother-in-law and sister-in-law

Why? Because:

• eniştem = my brother-in-law
• yengem = my sister-in-law
• yeğenim = my nephew/niece

So the sentence is being told from the speaker's point of view, and context strongly suggests the speaker was the one hosting.

It can mean either.

Turkish yeğen is gender-neutral:

• yeğenim = my nephew / my niece

Turkish often does not mark gender in nouns the way English does. You usually figure it out from context if it matters.

başucundaki means something like the one at the bedside / the one near the head side.

It is built from:

• baş ucu = head side, bedside
• başucunda = at the bedside / by the head side
• başucundaki = the one that is at the bedside

The ending -daki/-deki turns a location phrase into something that modifies a noun.

So:

• masadaki kitap = the book on the table
• odadaki çocuk = the child in the room
• başucundaki oyuncak = the toy at the bedside

oyuncağını breaks down as:

• oyuncak = toy
• -ı / -ın / -ı type endings here show possession and accusative
• because of sound changes, oyuncak becomes oyuncağ-

Most naturally in this sentence:

• oyuncağını = his/her toy
  • accusative

So it is:

• oyuncak
  • -ı (his/her) + -nı (accusative)
• written as oyuncağını

Two things to notice:

1. k → ğ
   Many Turkish words ending in k soften before a vowel:

   • oyuncak → oyuncağı

2. buffer n
   Turkish often inserts n between a possessive ending and a case ending:

   • oyuncağı
     • -ı → oyuncağını

In isolation, yes, it can be ambiguous.

Depending on context, oyuncağını could be understood as:

• his/her toy (3rd person possessed, accusative) or
• your toy (2nd person singular possessed, accusative)

But in this sentence, the natural reading is his/her toy, meaning the nephew's/niece's own toy.

Context is what decides it.

Because Turkish does not always state the possessor if it is obvious from context.

So başucundaki oyuncağını most naturally means:

• the toy at his/her bedside

The sentence does not explicitly say its bedside with a separate possessive phrase, but listeners understand it from the subject yeğenim.

This kind of omission is very normal in Turkish when the meaning is clear.

This is a very common Turkish pattern meaning:

• couldn't help showing
• couldn't resist showing
• couldn't refrain from showing

It is formed with:

• verb stem: göster- = show
• -meden = without doing
• edemedi = could not do

Literally, it looks like:

• he/she could not do without showing

But idiomatically, it means:

• he/she just had to show it
• he/she couldn't help showing it

Similar examples:

• gülmeden edemedi = he/she couldn't help laughing
• bakmadan edemedim = I couldn't help looking
• sormadan edemeyeceğim = I won't be able to help asking

Because this is an idiomatic Turkish construction.

• -meden = without doing
• edememek = to be unable to do

Together, they create the sense:

• not being able to avoid doing something

So although there is negative grammar inside it, the final meaning is not simply negative. It means the action happened because the person could not resist it.

English has similar idea patterns, even if the grammar is different:

• I couldn't help laughing
• She couldn't resist telling him

Because both are specific direct objects.

In Turkish, the accusative often marks a direct object that is:

• definite
• specific
• identifiable in context

So:

• yengemi = my sister-in-law, as a specific person
• oyuncağını = his/her toy, as a specific toy

Compare:

• oyuncak gösterdi = showed a toy / showed toys
• oyuncağını gösterdi = showed his/her toy

That accusative helps signal that the object is a particular one.

Because the verb is historically etmek, but in many forms it appears as ed- before vowel-initial endings.

So:

• dictionary form: etmek
• but: edemedi, ediyor, eder

This is normal:

• yardım etmek → yardım ediyorum
• fark etmek → fark ettim but fark ediyor

So edemedi is the expected form.

The sentence follows a very typical Turkish pattern:

• time
• background clause
• subject
• object
• main verb

So:

• Cumartesi akşamı = time
• eniştemle yengemi ağırlarken = background / while-clause
• yeğenim = subject
• başucundaki oyuncağını = object
• göstermeden edemedi = main verb

Turkish usually puts the main verb at the end, and everything before it builds the context step by step. That is why long Turkish sentences can feel backwards to English speakers at first.