What RPG do you play in which characters die as frequently (and swiftly) as they would if they actually fought giants, dragons, and purple worms?
Depends on what exactly you're asking.
Lethality and actions having consequences isn't necessarily synonymous with characters dying.
For example, even if I dial up some of the optional GURPS rules that make things harsher, that doesn't necessarily mean that PCs die frequently. It means that the approach to encounters is different.
Rather than a PC going toe-to-toe with a dragon, a dragon fight may instead be more like how battling Smaug is described in The Hobbit: he was an encounter formidable enough to take on a small militia.
He was felled by an arrow due to a secret vulnerability being known to a specially trained character using an arrow made for that purpose. Acquiring that information could itself be part of solving the encounter.
So, instead of a high level fighter having buckets of HP, they may instead have followers or sidekicks; access to esoteric information; or a special item. Rather than the dragon being an average encounter, it's the endpoint of an adventure's journey.
To more directly answer your question, I'll say GURPS*, DCC (at lower levels,) and some OSR games.
*as a modular system, you could also use the optional rules that make the game more akin to contemporary D&D or super heroes if you prefer that
I'll also add that, while not dragons, a FFG Star Wars character being hit by weapons mounted on a ship could mean instant death or severe injury very quickly. Even as a narrative-driven system, it acknowledges that certain types of injury are catastrophic.