When you arrive at Outskirt Stable, you’ll meet a man called Toffa. Speak to him, and he’ll tell you the story of the white stallion — and give you the Royal White Stallion side quest.
From Outskirt Stable, take the road north, turning left at the first intersection. Cross Manhala Bridge and follow the road right, around and up a hill. Dismount and look around. The wild white stallion is in this area. (Browse the gallery above for maps that show you the way.)
If you haven't upgraded your stamina, you won’t be able to catch the white stallion. Also, even though all horses are skittish in Breath of the Wild, this one is terrified of everyone. She’ll bolt at the first sign of company. To sneak up on her, we ate and drank stamina-boosting food and elixirs.
You’ll need stamina upgrades to calm the white stallion, too. You know that thing you do when you get on a while horse, where you mash on L to calm them down? That takes an extra long time with the white stallion, and if you don’t have stamina upgrades (we had four when we finally tamed her), she’ll buck you right off.
When you’ve caught her, return to Outskirt Stable, register and name your horse, and speak to Toffa again. Not only will you have a horse with high stamina, but you’ll also receive the royal bridal and the royal saddle, both of which you can see above in the customization section’s gallery.