Kızım arkadaşına mesaj attı ve onlar hemen barıştı.

It is kız + -ım.

• kız = daughter / girl
• -ım = my

So kızım means my daughter here.

The suffix changes by vowel harmony, so you may also see forms like evim (my house) or annem (my mother).

Because arkadaşına means to his/her friend, not just to a/the friend.

It breaks down like this:

• arkadaş = friend
• -ı = his/her
• -na = to

So arkadaşına = to her friend in this sentence.

Turkish often leaves the possessor unstated when it is obvious from context. Since the subject is kızım, arkadaşına is naturally understood as to her friend.

Compare:

• arkadaşa = to a/the friend
• arkadaşına = to his/her friend

That n is a buffer consonant.

When a noun already has the 3rd-person possessive suffix (-ı / -i / -u / -ü) and then another ending is added, Turkish often inserts n to connect them smoothly.

So:

• arkadaşı = his/her friend
• arkadaşına = to his/her friend

This same pattern appears in many words:

• evi = his/her house
• evine = to his/her house

Yes, atmak literally means to throw, but in Turkish it is also used in common expressions.

mesaj atmak is a very common everyday way to say to send a message / text.

So mesaj attı means sent a message.

Another possible verb is göndermek:

• mesaj gönderdi = sent a message

But mesaj attı is very natural in spoken Turkish, especially for texting or messaging.

Because Turkish normally prefers Subject–Object–Verb word order.

So instead of English-style My daughter sent a message, Turkish naturally says something more like:

• My daughter / to her friend / message / sent

That is why you get:

• Kızım arkadaşına mesaj attı
• onlar hemen barıştı

The verb often comes last in a neutral sentence.

They are in the simple past (often called the -DI past).

• attı = sent
• barıştı = made up / reconciled

This tense is used for completed past actions, especially when the speaker presents them as a definite fact.

The ending changes by vowel harmony and consonant harmony:

• geldi = came
• gitti = went
• yaptı = did
• attı = threw / sent

Yes, it could be left out.

Turkish often drops subject pronouns because the context already makes the subject clear. But onlar can be included for:

• clarity
• emphasis
• contrast

Here it helps make it clear that the two people later made up.

So all of these can work, depending on style and context:

• ve onlar hemen barıştı
• ve hemen barıştılar

With an overt plural subject like onlar, Turkish often allows the verb to stay in the singular form in everyday usage.

So both of these are possible:

• Onlar hemen barıştı
• Onlar hemen barıştılar

The version with barıştı is very natural. The version with barıştılar is also correct and may sound a bit more explicit.

A useful thing to remember:

• If onlar is present, singular or plural verb agreement can both occur.
• If onlar is omitted, barıştılar is usually clearer if you mean they.

It comes from barışmak, which means to make peace, to reconcile, or to make up.

It is not a direct-object verb like English to fix something. It already contains the idea of two sides becoming friendly again.

So:

• barıştı = he/she made peace / made up
• barıştılar = they made up

You can also say who someone reconciled with by using ile:

• Arkadaşıyla barıştı = She made up with her friend

In your sentence, the two people are already understood from context, so no extra phrase is needed.