Başın sağ olsun, çok üzüldüm.

Because it is an old fixed condolence expression. Word for word, it is something like may your head be healthy/well.

That literal image does not match how English expresses condolences, so it is best learned as an idiom. In real use, Turkish speakers understand it simply as a standard, natural way to say my condolences or I’m sorry for your loss.

It is built into the noun.

• baş = head
• başın = your head

The ending -ın is the 2nd person singular possessive suffix, meaning your.

So Turkish often does not need a separate word like English your. The possession is shown by the ending.

Olsun comes from olmak (to be / to become).

Here it means something like:

• let it be
• may it be

So:

• başın sağ olsun = literally may your head be well/healthy

Grammatically, olsun is a 3rd person wish/imperative-like form, because the thing that is supposed to be healthy is başın (your head), not you directly.

Because they are different structures.

• sağ ol = be well / stay healthy
  This is said directly to the person. It is also commonly used like thanks.
• başın sağ olsun = may your head be well
  This is the condolence expression.

So sağ ol and başın sağ olsun are related in wording, but they are not interchangeable.

This is very normal in Turkish.

• üzüldüm literally means I became sad / I was saddened
• In context, it often means I was very sorry to hear that or I’m deeply sorry

So even though it uses past tense, it can express a current emotional reaction to news you just heard.

A close alternative is:

• Çok üzgünüm = I am very sad / I am very sorry

Both are natural, but çok üzüldüm often sounds like hearing this affected me deeply.

Yes, başın is singular informal (your, to one person you address as sen).

For polite singular or plural, use:

• Başınız sağ olsun

So:

• to a friend, sibling, close acquaintance: Başın sağ olsun
• to an older person, someone you address formally, or multiple people: Başınız sağ olsun

Mostly, yes.

Başın sağ olsun is strongly associated with death and bereavement. It is the normal thing to say when someone has lost a relative or loved one.

For ordinary bad news, it would usually sound too strong or inappropriate. In those situations, Turkish would use other expressions instead.

So it is safest to use başın sağ olsun / başınız sağ olsun for loss through death.

Başın sağ olsun is enough by itself.

Adding çok üzüldüm makes it warmer and more personal:

• Başın sağ olsun. = standard condolence
• Başın sağ olsun, çok üzüldüm. = standard condolence + personal sadness

You can say either one, depending on how much you want to say.

Because Turkish verbs already show the subject.

In üzüldüm, the ending -düm tells you it means I:

• üzüldüm = I was saddened / I got sad

So ben is not necessary. You could say Ben çok üzüldüm, but that usually adds emphasis to I.

Yes, Turkish word order is flexible, and that version is understandable.

But Başın sağ olsun, çok üzüldüm sounds very natural because it starts with the actual condolence formula, then adds the personal reaction.

Also, many speakers would say them as two separate sentences:

• Başın sağ olsun. Çok üzüldüm.

That is just as natural.

Roughly like bah-shuhn.

A few helpful notes:

• ş = sh
• the ı in başın is the Turkish ı, a vowel that does not exist in English
• it is not the same as English ee or ih

So:

• baş ≈ bahsh
• başın ≈ bah-shuhn (approximate English guide only)

The exact Turkish ı takes some practice, but learners are usually understood even before they perfect it.