Learn Hardcore Turkish: Otobüste cüzdanını kaybeden adam çok endişeliydi. - The man who lost his wallet on the bus was very worried.

So otobüste means “on the bus / in the bus”.
Turkish usually doesn’t use separate prepositions like in, on, at; it uses case endings instead.

What does cüzdanını mean, and why doesn’t it just say cüzdan?

Cüzdanını is not just “wallet”; it’s “his/her wallet” as a direct object.

Breakdown:

cüzdan = wallet
cüzdanı = wallet + -ı (3rd person possessive: his/her wallet)
cüzdanını = cüzdan-ı + -nı (accusative case: specific direct object)

So:

cüzdanı = his/her wallet (no case marking)
cüzdanını = his/her wallet (as the thing that is lost)

In cüzdanını kaybeden adam:

cüzdanını = his wallet (object)
kaybeden = who lost
adam = the man

So it describes “the man who lost his wallet”.
The owner (his) is understood from context (it’s the man), so Turkish doesn’t need to say onun cüzdanını (“his wallet”) unless it wants extra clarity or contrast.

Why is it kaybeden and not kaybetti?

Kaybeden is a participle, not a normal past tense form.

kaybetti = (he) lost (a complete verb in a full sentence)
kaybeden = who loses / who lost (a verb turned into an adjective)

In Turkish, the suffix -en / -an makes a “who/that VERB-ed” form:

gülen çocuk = the child who is laughing / who laughed
konuşan kadın = the woman who is speaking / who spoke
cüzdanını kaybeden adam = the man who lost his wallet

So kaybeden is needed because it’s modifying “adam” like an adjective.
If you used kaybetti, you’d need to make a full sentence:

Adam otobüste cüzdanını kaybetti. = The man lost his wallet on the bus.

How does the phrase “cüzdanını kaybeden adam” work grammatically?

It’s a relative clause: a mini-sentence that describes adam.

Literally:

[cüzdanını kaybeden] adam
= the man [who lost his wallet]

Structure inside:

(adam) – the subject (not written, understood from context)
cüzdanını – “his wallet” (object)
kaybeden – “who lost” (participle)

In Turkish, everything that describes the noun comes before that noun:

English: the man [who lost his wallet]
Turkish: [cüzdanını kaybeden] adam

So “cüzdanını kaybeden” is functioning as a big adjective attached to adam.

Could “cüzdanını kaybeden adam” mean “the man whose wallet was lost,” instead of “the man who lost his wallet”?

In normal interpretation, no.
“Cüzdanını kaybeden adam” is understood as:

the man who (himself) lost his wallet

Inside the relative clause, the unspoken subject is adam, so:

adam cüzdanını kaybetti → “the man lost his wallet”
becomes
cüzdanını kaybeden adam → “the man who lost his wallet”

If you wanted “the man whose wallet was lost”, you would phrase it differently, e.g.:

Cüzdanı kaybolan adam = the man whose wallet got lost
Cüzdanı çalınan adam = the man whose wallet was stolen

Why does endişeliydi end with -ydi? Is that the past tense?

Yes. Endişeliydi is “was worried”.

Breakdown:

endişeli = worried (adjective)
idi = was (past tense of “to be”)
endişeli + idi → endişeliydi (they fuse in normal speech and writing)

So:

adam endişeli = the man is worried
adam endişeliydi = the man was worried

In your sentence:

adam çok endişeliydi = the man was very worried

Can I say “adam çok endişeliydi” instead of putting adam earlier in the sentence?

Yes. In fact, “Otobüste cüzdanını kaybeden adam çok endişeliydi” already has that order:

Subject: Otobüste cüzdanını kaybeden adam
Predicate: çok endişeliydi

You could move parts around for emphasis:

Adam otobüste cüzdanını kaybeden çok endişeliydi. → sounds wrong (splits the structure)
Otobüste cüzdanını kaybeden adam çok endişeliydi. → natural
Otobüste cüzdanını kaybeden adam çok endişeliydi. (as given) → subject first, then “was very worried”

In Turkish, the relative clause must stay together in front of the noun it describes, so you don’t separate kaybeden from adam.

Is çok necessary? What’s the difference between “adam endişeliydi” and “adam çok endişeliydi”?

Çok means “very, a lot” here.

adam endişeliydi = the man was worried
adam çok endişeliydi = the man was very worried / extremely worried

You can safely omit çok; the sentence will still be correct:

Otobüste cüzdanını kaybeden adam endişeliydi.

It just sounds a bit weaker emotionally than çok endişeliydi.

How would I say “The man who lost his wallet on the bus is very worried (now)” instead of “was very worried”?

You just remove the past tense ending -ydi and keep the adjective in the present:

Otobüste cüzdanını kaybeden adam çok endişeli.
= The man who lost his wallet on the bus is very worried.

Compare:

… çok endişeli. → is very worried (present)
… çok endişeliydi. → was very worried (past)

How would I say “the wallet that the man lost on the bus,” using a similar structure?

Here, “the wallet” is the thing that was lost, and “the man” is just extra information.
You would usually make a relative clause where “wallet” is the noun being described:

Otobüste adamın kaybettiği cüzdan
= the wallet that the man lost on the bus

Breakdown:

adamın = the man’s
kaybettiği = that he lost
(from kaybetmek
- -tiği → “that he lost”)
cüzdan = wallet

So the pattern switches:

cüzdanını kaybeden adam = the man who lost his wallet
adamın kaybettiği cüzdan = the wallet that the man lost