Battaniyeyi bebeğin üstüne koydum.

Because it is the direct object of the verb koymak (to put) and it is a specific/definite object: the blanket, not just a blanket.

• battaniye = blanket
• -yi = accusative ending

So:

• battaniye = a blanket / blanket
• battaniyeyi = the blanket

The extra y is a buffer consonant added because the noun ends in a vowel and the case ending also begins with a vowel.

Because Turkish uses the genitive case here.

• bebek = baby
• bebeğin = of the baby / the baby's

This is used because üstüne is part of a structure that works like the baby’s top/on top of the baby.

So the phrase is built like this:

• bebeğin üstüne
• literally: to the baby’s upper side
• natural English: onto/on the baby

This kind of structure is very common in Turkish with words like:

• üst = top
• alt = bottom/under
• iç = inside
• ön = front
• arka = back

This is a common sound change in Turkish.

Some words ending in k change that k to ğ when a vowel-initial suffix is added.

So:

• bebek
  • -in → bebeğin

This is called consonant softening.

Other common examples:

• çocuk → çocuğun
• renk → rengin
• yürek → yüreği

So bebeğin is the normal correct form.

Üstüne means onto it / on top of it / over it, depending on context.

It breaks down like this:

• üst = upper part, top
• -ü = its
• -ne = to / onto

So üstüne literally means something like onto its top.

In bebeğin üstüne, the full sense is:

• bebeğin üstüne = onto the baby / on top of the baby / over the baby

In this sentence, English would usually say I put the blanket over the baby, but Turkish expresses it with this üstüne structure.

Because üst here behaves like a possessed noun.

Turkish often uses body-part/location-type words in possessed forms:

• üstü = its top / upper part
• altı = its bottom / underside
• içi = its inside

So:

• bebeğin üstü = the baby’s top / the top of the baby
• bebeğin üstüne = onto the baby / onto the baby’s upper side

This may feel strange to an English speaker, but it is a very normal Turkish pattern.

Because -e / -a and -de / -da express different ideas.

• üstüne = onto / to the top of → movement toward a place
• üstünde = on / on top of → location, already there

Since koymak means to put/place, it usually goes with a destination, so üstüne fits.

Compare:

• Battaniyeyi bebeğin üstüne koydum.
  I put the blanket onto/over the baby.

• Battaniye bebeğin üstünde.
  The blanket is on the baby.

Koydum means I put.

It breaks down like this:

• koy- = verb stem, put/place
• -du- = past tense
• -m = I

So:

• koydum = I put

Turkish often includes the subject inside the verb ending, so you do not need a separate word for I.

Because the verb already tells you the subject.

In koydum, the ending -m means I.

So Turkish usually does not need ben unless you want emphasis.

• koydum = I put
• ben koydum = I put it / I was the one who put it

This is very common in Turkish.

Yes, it is very natural.

Turkish often puts the verb at the end, and a common order is:

object + place/direction + verb

So here:

• Battaniyeyi = the blanket
• bebeğin üstüne = onto/over the baby
• koydum = I put

That said, Turkish word order is flexible. Different orders can add emphasis.

For example:

• Bebeğin üstüne battaniyeyi koydum.
  More emphasis on where it was put.

But the original sentence is completely natural and neutral.

Turkish often uses noun + case ending structures where English uses prepositions.

English says:

• over the baby
• on the table
• in the room

Turkish often says things more like:

• the baby’s upper side
• the table’s top
• the room’s inside

So instead of a single preposition like over, Turkish uses a phrase such as:

• bebeğin üstüne

This is one of the big differences between English and Turkish structure.

Yes, grammatically it can. The phrase bebeğin üstüne literally does mean onto/on top of the baby.

But in normal context, with blanket, the natural interpretation is usually:

• I put the blanket over the baby

So the exact English translation depends on context, but the Turkish wording is perfectly normal.

Turkish does not have a separate word exactly like English the.

Instead, definiteness is often shown in other ways, especially by the accusative case for direct objects.

That is why:

• battaniye koydum can suggest I put a blanket
• battaniyeyi koydum suggests I put the blanket

So the ending helps do some of the work that the does in English.

Usually that would sound odd unless there is a special context.

• bir battaniye = a blanket
• battaniyeyi = the blanket

Using bir and the accusative together is possible in some contexts, but it usually adds a special nuance, not just simple a blanket.

If you mean I put a blanket over the baby, the more natural sentence is:

• Bebeğin üstüne bir battaniye koydum.

If you mean I put the blanket over the baby, then:

• Battaniyeyi bebeğin üstüne koydum.

In modern Turkish, ğ usually is not pronounced like a strong consonant in the middle of a word.

In bebeğin, it mostly lengthens or smooths the vowel before it.

So it sounds roughly like:

• be-be-in
  or
• be-bee-in with a slight glide

Do not try to pronounce it like a hard g.

Here it is word by word:

• Battaniyeyi
  battaniye

  • -yi
    blanket
    • definite accusative
      = the blanket

• bebeğin
  bebek

  • -in
    baby
    • genitive
      = of the baby / the baby’s

• üstüne
  üst

  • -ü
    • -ne
      top/upper part
      • its
        • to/onto
          = onto it / on top of it

• koydum
  koy-

  • -du
    • -m
      put
      • past + I
        = I put

So very literally, the sentence is something like:

• I put the blanket onto the baby’s upper side.

Natural English:

• I put the blanket over the baby.