First of all: in my experience, unless you truly are a nasty DM, the risk of dying in D&D 5E is almost negligible. There might be other issue with encounter balancing if you expect them to die so much, but this is not the place to discuss that. In any case, aside from agreeing with almost every other answer that you received (for instance this and this), I wanted to add an extra suggestion:
Tl;dr: I propose to adapt the Recoveries from 13th Age to 5E, which in some sense are similar to enhancing the power of the standard Hit Dice in order to allow them to cure more and to be used in combat as well.
If you are really afraid of killing your players to easily, you can borrow the Recovery mechanics from 13th Age, which allows the game to run almost as smoothly even without any healing character. References from that game are available here, specifically at the section on Rally and Recoveries
In short, the idea is to introduce a pool of points (a fixed amount, let's say 8, more on this later) which we call Recoveries. Each Recovery point allows a character to heal themself by 1/4 of their hit point maximum. The Recoveries can be spent in two ways:
- In combat, one can spend one recovery as a standard action. You can use in this way only oncr per battle, though.
- Out of combat, a character can spend any amount of recovery to heal as a short rest.
Interaction with the rest of the edition
Depending on how much you want to tinker the system, a lot of features can interact with these mechanics, for instance:
- Healing potions can heal you of one Recovery instead of giving you a "free healing". This way potions scale up in an automatic way, but as is they are not different than using a Recovery alone. So instead they can provide an extra benefit: remove also a condition, heal an extra amount or require another action to be used (i. e. a bonus action) or that they can be used on other character (thus allowing to cure someone who is at 0 hp).
- The Fighter ability second wind can spend a recovery instead of a "free healing", but it can be used as a bonus action. Moreover, to compensate losing the free healing, the Fighter might gain an extra recovery and/or become able of using two per combat.
- Similar things might be implemented for spells and such, I think you get the idea.
- Of course these mechanics should replace the mechanics of the hit dices for out of combat healing.
How many Recoveries?
In 13th Age each adventuring "day" lasts a fixed number of encounters: all the mathematics of the game is rigorously balanced to fight in 4 encounters between each long rest. This allows you to compute almost exactly the number of Recoveries needed to maintain the equilibrium.
In 5E the number of encounter is not fixed: the suggested one is between 6 and 8, depending on the challenge. When I prepare my campaigns, I try to hover toward the lower end of this range (4-6) in order to allow myself to increase the challenge. In any case, I would say that around 2 Recoveries for each encounter can allow you to get a you a rough idea. But There is a caveat on this reasoning: damage dealt by monsters in 5E is really tuned down! The vast majority of the enemies in the game are more of the tanky type. Given this, I would give each character around 1.5 Recoveries per encounter, thus if you plan on doing 6 encounter per day, around 9 Recoveries should suffice.
Moreover, another caveat is how much you want the Recoveries to interact with the other forms of healing: if Rrcoveries are the only means by which one can cure oneself, maybe you can increase that number or or lower it in the opposite case.
Edit: As @medix2 commented, there is something similar in the Dungeon Master guide, called Healing Surge, that you can find discussed here. Personally, I don't like that variant too much but only because it is somewhat verbose in game: you start rolling, you decide if you want to roll another one, then you have to keep track of how many you spent and how many you got back and so on. The 13th Age rule that I showed is very straightforward: you have one pool, it replenishes in only one way (long rest) and every time you use it the same thing happens.