In Sekiro: Shadows Die Twice, healing potions are called Healing Gourd. These items come in very limited quantity. Check this page to learn how to carry more Healing Gourds?
You get a Healing Gourd from Kuro - this happens in the prologue. Sadly, the default number of these healing potions is one, meaning that you can only heal once.
After using a Healing Gourd, you are unable to use it again until you go back to a Sculptor's Idol ("bonfires") and Rest. This option resupplies all consumables. However, the downside is that all enemies respawn. You can't go back to a Sculptor's Idol when you are in the middle of a boss fight - you need to win battles with only a single Healing Gourd.
The game allows you to increase the number of carried Healing Gourds. There are a few moments in the game where you can get a Gourd Seed - you can find them during exploration or receive after you beat a boss. Check the Gourd Seeds page to learn more about them.
Take Gourd Seeds to Dilapidated Temple and meet with Emma the Physician. Give her this item - your Healing Gourds will increase by one.
Bring Gourd Seeds to Emma as soon as you get them. Being able to heal yourself more times will make the upcoming battles easier.