Paulnamid's answer covers your mistake about Gilgamesh, and briefly mentions what you've got wrong about Artoria. But to go into more detail on Artoria...
Artoria has the heart of a dragon. Or, depending on who you ask, her heart is a dragon. She can as such store and produce massive amounts of prana all on her own, and does so as naturally as (and via) breathing. If a mage is a minor machine at this, she's a massive factory. The problem for her is that she needs something of a starter supply to activate the core, and it's this she was originally in short supply of in F/SN. She starts out with no supply of energy from Shirou, at all. So there's no issue with Shirou withstanding her Mana Bursts, as he was never involved in that in any way. She was initially relying entirely on her stored reserves, and used up something like a fifth of what she started with in her first encounter with Berserker. But eventually, in the suitable route, Shirou gets his circuits connected to Artoria's with Rin's help, and her draconic magic core activates (well enough for her to achieve what she does for the rest of the war, anyway)*. Depending on which version of the story you have digested, you might see Shirou encounter a dragon in this sequence, and the core light up with a prismatic light as it activates with Shirou's energy. After this point, Shirou is essentially only providing the fuel necessary to keep Artoria's core running, and the core's production covers her mana burst.**
The distinction then is that, as a Servant, Karna has no intrinsic ability to produce his own prana supply, at least not at the level necessary to sustain everything he can do. Artoria, however, has a prana factory intrinsic to her very being (though it may need a jump start).
*In the Unlimited Blade Works route, Shirou is instead connected to Rin, who has a heart of dolphins or something. Rin then fuels both Artoria and Shirou.
**You may be asking yourself, "why doesn't Artoria just feed her own core the energy needed to kickstart it? She's got a lot more than Shirou does for most of this time." The answer there doesn't seem to get spelled out as explicitly, but the short of the matter is that she was never trained in any of the magecraft arts, and she has no real understanding about how to use her core and her absolutely amazing draconic magic circuits properly. But if she had been... So she has difficulties really activating the core on her own, and it's the direct hookup of her core to Shirou's supply that enables it to finally get going.