This page explains how to start and complete the Good-Sized Horse side quest.
How to Start the Quest
How to Complete the Quest
Bring Zyle a horse that you do not want to keep. It can be a wild horse or one of the horses that you have registered with the stables. You will never see this horse again, so be sure that you really want to get rid of it. Agree to sell Zyle the horse and he will give you 300 rupees.