Learn Turkish: Bize yardım teklif ettiler.

In Turkish, compound verbs formed with etmek or olmak are written together as a single word only if the noun undergoes a sound change, such as a dropped vowel or a doubled consonant. The Arabic loanword teklif does not lose a vowel or double its final consonant when combined, so teklif etmek is always written as two separate words.

Why is bize in the dative case here?

The compound verb teklif etmek (to offer) requires the dative case for the person receiving the offer. Just as you offer something "to" someone in English, here they offered help to us, which translates to the dative pronoun bize.

Can I say yardım etmeyi teklif ettiler instead of yardım teklif ettiler?

Yes, you can. Yardım teklif ettiler translates neatly to "they offered help" using yardım simply as a noun. If you use yardım etmeyi teklif ettiler, you are literally saying "they offered to help" by using the verbal noun etmeyi in the accusative case. Both are perfectly natural and convey the same meaning.

Why isn't yardım in the accusative case, since it is the thing being offered?

In this sentence, yardım is indefinite, meaning they offered "help" in general, not a specific or previously mentioned help. When a direct object is non-specific, it does not take the accusative suffix and typically sits right next to the verb.