Küçük kız gülünce yanakları daha da güzelleşiyor.

It has two parts:

• Küçük kız gülünce = when the little girl smiles/laughs
• yanakları daha da güzelleşiyor = her cheeks become even more beautiful

So the pattern is:

• [when-clause] + [main clause]

This is very common in Turkish.

Gülünce means when she smiles/laughs or once she smiles/laughs.

It comes from:

• gülmek = to smile / to laugh
• -ünce / -ınca / -ince / -unca = a suffix meaning when or once

So:

• gül-
  • -ünce → gülünce = when (someone) smiles/laughs

Because of vowel harmony.

The verb stem is gül-, and the vowel in the stem is ü. Turkish chooses the form of the suffix to match that vowel, so you get:

• gülünce

Compare:

• gelince = when he/she comes
• bakınca = when he/she looks
• oturunca = when he/she sits

The exact shape of the suffix changes to fit the vowels in the word.

It can mean both, depending on context.

• In some contexts, gülmek = to laugh
• In others, it can mean to smile

In this sentence, because it talks about the little girl’s cheeks becoming prettier, English would often use smiles more naturally.

The subject of güzelleşiyor is yanakları = her cheeks.

A learner might first think küçük kız is the subject of the whole sentence, but actually:

• küçük kız is the subject of gülünce
• yanakları is the subject of güzelleşiyor

So the sentence means:

• When the little girl smiles, her cheeks become even more beautiful.

Not:

• When the little girl smiles, she becomes more beautiful
  That would be a different sentence.

Because yanakları here means her cheeks.

Breakdown:

• yanak = cheek
• yanaklar = cheeks
• yanakları = his/her cheeks (or in some contexts their cheeks)

So the -ı here is a possessive ending, not just a random extra ending.

In this sentence, it means the cheeks belong to the little girl.

It could, but Turkish often leaves out information that is already clear from context.

Since the sentence already mentioned küçük kız in the first part, Turkish can naturally continue with just:

• yanakları = her cheeks

So:

• Küçük kız gülünce yanakları daha da güzelleşiyor.

is natural and efficient.

If you said:

• Küçük kız gülünce küçük kızın yanakları daha da güzelleşiyor

that would sound repetitive unless you wanted special emphasis or extra clarity.

Daha da means even more.

• daha = more
• da here adds emphasis, giving the sense of even

So:

• daha güzel = more beautiful
• daha da güzel = even more beautiful

In this sentence:

• daha da güzelleşiyor = becomes even more beautiful

It can be broken down like this:

• güzel = beautiful
• -leş- / -laş- = become, turn into
• -iyor = present continuous / progressive

So:

• güzelleşiyor = is becoming beautiful
• more naturally in English here: is getting prettier / becomes more beautiful

This -leş- / -laş- pattern is very useful in Turkish:

• temiz = clean → temizleşmek = to become clean
• zor = difficult → zorlaşmak = to become difficult
• güzel = beautiful → güzelleşmek = to become beautiful

Not necessarily only right now.

Turkish often uses -iyor not just for something happening at this exact moment, but also for:

• a general observation
• a repeated situation
• a vivid, natural description

So here, güzelleşiyor can mean something like:

• her cheeks get even prettier when she smiles

Even though the form is progressive, the meaning in English may sound more natural as a general truth.

Because Turkish does not always show plural agreement on the verb the way English does.

With a third-person plural subject, especially a non-human one, a singular verb form is very normal.

So:

• yanakları güzelleşiyor = natural Turkish

You do not need a special plural verb ending here.

Yes, Turkish word order is flexible, but this order is the most natural and clear.

Current order:

• Küçük kız gülünce yanakları daha da güzelleşiyor.

This puts the when-clause first, which is very common.

You can move things around in Turkish for emphasis, but not every reordered version sounds equally natural. For a learner, this sentence order is a very good model to follow.

It is the same particle in form, but here its effect is different.

Usually:

• Ben de = me too
• O da geldi = he/she came too

But in combinations like daha da, it works as an intensifier:

• daha da iyi = even better
• daha da kötü = even worse
• daha da güzelleşiyor = becomes even more beautiful

So in this sentence, it does not mean also in the usual sense. It strengthens daha.