I think a better translation for you to understand this might be: I read a newspaper to him.
Would you be able to drop 'Ben' here? (I'm assuming it would sound strange, if not completely incorrect).
Can you please explain how this sentence is correct
- Ben -- I -- nominative case for subject
- ona -- to him/her -- dative case for "recipient"
- bir gazete -- a newspaper -- nominative case, since it's the direct object but it is indefinite
- okurum = oku-r-um, from oku-mak (to read), -r- (sign of the aorist tense), -um (personal ending for "I") = I read
I read a newspaper to him/her.