Questions & Answers about 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.