Kuzenim, Haziran sonunda sahile gidersek o mavi mayoyu giymek istiyor.

Because kuzenim means my cousin.

• kuzen = cousin
• -im = my

So:

• kuzen = cousin
• kuzenim = my cousin

Turkish often shows possession directly on the noun instead of using a separate word like my.

Here, Kuzenim is being used as the topic of the sentence, almost like setting up As for my cousin...

So the sentence structure is roughly:

• Kuzenim, ... istiyor.
• My cousin, ... wants ...

The comma is not always required in all Turkish writing, but it can help show that kuzenim is being set off as the topic.

It means at the end of June.

Breakdown:

• Haziran = June
• son = end
• sonunda = at/on its end, at the end of

So Haziran sonunda literally means something like at June’s end.

This is a very common Turkish time expression.

Because sahile means to the beach.

• sahil = beach / shore / seaside
• sahil-e = to the beach

The ending -e / -a is the dative case, often used for motion toward something.

Compare:

• sahil = the beach
• sahilde = at/on the beach
• sahile = to the beach

Since the idea is go to the beach, Turkish uses sahile.

Gidersek means if we go.

Breakdown:

• gitmek = to go
• its aorist/future-like stem here becomes gider-
• -se / -sa = if
• -k = we

So:

• gidersek = if we go

This is a very common way to form conditional clauses in Turkish.

This is because the conditional here is built on the aorist-type form of the verb:

• gitmek → gider
• gider
  • -sek → gidersek

Many common Turkish verbs have stems that change in certain forms. Gitmek is one of them.

So even though the dictionary form is gitmek, you will often see forms like:

• giderim = I go / I would go
• giderse = if he/she goes
• gidersek = if we go

This is just something learners get used to through exposure.

The ending -sek clearly means if we go, so the subject of that clause is we.

That creates a sentence with two different subjects:

• gidersek = if we go
• kuzenim ... istiyor = my cousin wants

So the full idea is:

• If we go to the beach at the end of June, my cousin wants to wear that blue swimsuit.

This kind of subject shift is completely normal in Turkish.

Because mayoyu is the definite direct object form.

• mayo = swimsuit
• mayoyu = the swimsuit

In Turkish, when the direct object is specific/definite, it usually takes the accusative ending.

Here the swimsuit is specific because it is identified as:

• o mavi mayo = that blue swimsuit

So Turkish uses the accusative:

• o mavi mayoyu

The -y- is a buffer consonant added because the noun ends in a vowel.

O means that here.

So:

• mavi mayo = blue swimsuit
• o mavi mayo = that blue swimsuit

It points to a specific item, which is one reason the object is definite and takes the accusative ending.

Turkish commonly expresses want to do something with:

• verb infinitive + istemek

So:

• giymek = to wear
• istiyor = wants
• giymek istiyor = wants to wear

This is very natural Turkish.

Similar examples:

• gitmek istiyorum = I want to go
• yemek istiyor = he/she wants to eat
• uyumak istiyoruz = we want to sleep

İstiyor is the present continuous form and is very commonly used in Turkish for present intentions, wishes, and current states.

• istiyor = wants / is wanting
• ister can mean something more like wants, would want, or a more general/habitual sense depending on context

In everyday Turkish, istiyor is the normal choice for wants in a sentence like this.

Good question. Turkish distinguishes this a bit differently than English.

• giymek = to put on / to wear
• giyiyor can mean is wearing depending on context

In this sentence, giymek istiyor means wants to wear / wants to put on that swimsuit.

So English may translate it as wants to wear, even though the Turkish verb can cover both the action of putting on and the state of wearing in different contexts.

Because Turkish normally places the main verb at the end.

So the structure is very often:

• topic / time / place / objects / other information + verb

In this sentence:

• Kuzenim = my cousin
• Haziran sonunda = at the end of June
• sahile gidersek = if we go to the beach
• o mavi mayoyu = that blue swimsuit
• giymek istiyor = wants to wear

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

Yes, to a degree. Turkish word order is flexible, especially for emphasis, although the default is still to keep the main verb near the end.

For example, the sentence could be rearranged to emphasize different parts, but:

• Kuzenim Haziran sonunda sahile gidersek o mavi mayoyu giymek istiyor.
• Haziran sonunda sahile gidersek kuzenim o mavi mayoyu giymek istiyor.

Both are understandable.

The original version uses Kuzenim as a topic at the start and sounds natural. Turkish often moves elements around for emphasis more freely than English does.