This page explains how to start and complete the Wild Horses side quest.
How to Start the Quest
Talk to Rensa at the Dueling Peaks stable, near Ha Dahamar Shrine, and talk to him again to accept the horse taming challenge.
Tame a Wild Horse in Two Minutes
When the countdown starts, go north to find a group of wild horses. Sneak up behind them (but not directly behind, or they may kick you), and mount the horse when you have the option to do so. Spotted horses are easier to tame than solid-color horses. For solid-color horses, you will have to tap the "soothe" button repeatedly until the horse calms down.
Talk to Rensa to complete the quest and get a purple rupee.