Learn Turkish: Kulübe yeni üyeler katıldı. - New members joined the club.

The verb katılmak (to join, to participate) always takes the dative case (-e/-a). Even though we say "join the club" in English, in Turkish you can think of it as members adding themselves to the club.

Does the p in kulüp change to a b because of the suffix?

Exactly. This is standard consonant mutation. When a word ends in a hard consonant like p and receives a suffix starting with a vowel (like the dative -e), it softens to b. So, kulüp becomes kulübe.

Is katılmak structurally related to the verb katmak?

Yes! Katmak means "to add." The -ıl suffix attaches to verbs to create a passive or reflexive meaning. So, katılmak literally means "to be added" or "to add oneself," which is why it translates perfectly to "to join."

Since the subject is the plural yeni üyeler, why is the verb singular (katıldı instead of katıldılar)?

When the subject of a sentence is a plural third-person (even for people), adding the plural suffix -lAr to the verb is optional. Saying katıldılar is grammatically fine, but leaving the verb in the singular form is very common and sounds perfectly natural.