This is the 670th most frequent Turkish word.
düzenlediği
"Organized by" or "arranged by."
Here, 'düzenlediği' modifies 'kitap okuma günleri' (reading days), explaining 'those which the library organized'.
Kütüphane, düzenlediği kitap okuma günleri sayesinde popüler oldu.
The library became popular thanks to the reading days it organized.
In this sentence, 'düzenlediği' is used as a participial verb form meaning 'that he/she organized' and acts as an adjective describing 'etkinlik' (event).
Ali, şirket adına bir etkinlik düzenlediği için övüldü.
Ali was praised for organizing an event on behalf of the company.
In this instance, 'düzenlediği' provides context to 'sergi' (exhibition), indicating the exhibition's organizer, enhancing the relation between the action and the subjects.