Onu baygın görünce tedirgin oldum.

A natural breakdown is:

• onu = him / her / it (object form of o)
• baygın = fainted, unconscious, passed out
• görünce = when/after seeing
• tedirgin oldum = I became worried / I got uneasy

So the sentence is literally something like:

• Seeing him/her unconscious, I became worried.
• Or more naturally: When I saw him/her passed out, I got worried.

Because onu is the object form.

• o = he / she / it
• onu = him / her / it

In this sentence, the speaker is seeing that person, so that person is the object of görmek (to see).

Compare:

• O baygın. = He/She is unconscious.
• Onu baygın gördüm. = I saw him/her unconscious.

So onu is required here because the person is what is being seen.

Görünce comes from:

• gör- = verb stem of görmek (to see)
• -ünce / -ınca / -ince / -unca = when / upon / after doing

Because of vowel harmony, it becomes -ünce here:

• görmek → görünce = when (someone) sees / when I saw / upon seeing

In this sentence, it means:

• when I saw
• upon seeing

So:

• Onu baygın görünce... = When I saw him/her unconscious...

The subject is understood from the main clause: I.

In Turkish, with this kind of -ince clause, the subject is usually the same as the subject of the main verb unless context makes it clear otherwise.

Here:

• tedirgin oldum = I became worried

So the natural interpretation is:

• When I saw him/her unconscious, I became worried.

That means I am the one who saw the person.

Because Turkish often leaves subjects unstated when the verb already shows who the subject is.

• oldum = I became

The ending -um tells you the subject is I.

So Turkish does not need ben here.

• (Ben) tedirgin oldum. = I became worried.

Adding ben is possible, but it is usually only used for emphasis or contrast.

Literally, tedirgin oldum means:

• I became uneasy
• I got worried
• I became anxious

Here:

• tedirgin = uneasy, worried, disturbed
• oldum = I became / I was

Turkish often uses olmak (to become / to be) with adjectives or states.

So:

• mutlu oldum = I became happy
• şaşırdım = I was surprised
• tedirgin oldum = I became uneasy

In English we might say I got worried, and Turkish expresses that with tedirgin oldum.

Because baygın is functioning as an adjective describing onu.

It means:

• unconscious
• fainted
• passed out

So:

• onu baygın görünce = when I saw him/her unconscious

This works similarly to English in:

• I saw him unconscious.

Turkish can place the adjective directly next to the object without adding anything extra.

They are close, but not exactly identical in feel.

• baygın = unconscious / in a fainted state
• bayılmış = has fainted / fainted

So:

• Onu baygın görünce focuses on the person's condition/state.
• Onu bayılmış görünce is generally less natural here.
• A more common alternative would be Onun bayıldığını görünce = When I saw that he/she had fainted

Your sentence is very natural and means the speaker saw the person in an unconscious state.

Yes, that is possible, and the meaning is very similar.

Compare:

• Onu baygın görünce tedirgin oldum.
• Onu baygın gördüğümde tedirgin oldum.

The first one with -ince often feels a bit more immediate and compact:

• When / upon seeing him/her unconscious, I got worried.

The second one with gördüğümde is also correct and means:

• When I saw him/her unconscious, I got worried.

So both are good, but görünce is shorter and very common in everyday Turkish.

Yes, Turkish word order is flexible, but the given version is very natural.

Standard order here is roughly:

• Onu baygın görünce = subordinate clause
• tedirgin oldum = main clause

This gives:

• When I saw him/her unconscious, I became worried.

You could rearrange it for emphasis, for example:

• Tedirgin oldum onu baygın görünce.

But this is less neutral. The original sentence is the most natural everyday phrasing.

Also, Turkish often puts the main verb at the end, which is exactly what happens with oldum.

It can mean either one.

Turkish third-person singular pronouns do not mark gender:

• o = he / she / it
• onu = him / her / it

So only context tells you whether the person is male or female.

Yes, learners sometimes notice that görünce looks similar to görünmek (to appear / to seem), but here it is from görmek (to see), not görünmek.

It is:

• gör-
  • -ünce → görünce

So this sentence does not mean:

• when he/she appeared faint

It means:

• when I saw him/her faint/unconscious

That is an important distinction.

The pattern is:

[object] + [descriptive word] + [verb + -ince] + [main clause]

So here:

• Onu = object
• baygın = description of the object
• görünce = when seeing
• tedirgin oldum = I became worried

This pattern is very common in Turkish:

• Onu mutlu görünce sevindim. = When I saw him/her happy, I was glad.
• Çocuğu ağlarken görünce üzüldüm. = When I saw the child crying, I got sad.
• Seni yorgun görünce endişelendim. = When I saw you tired, I got concerned.

So your sentence is a very useful model for similar expressions.