Artax was evidently profoundly affected by the Swamp of Sadness, eventually sinking into a deep depression. However, the Swamp had no effect whatever on Atreyu, which Artax thought was likely due to him wearing the Amulet. That being the case, why didn't Artax just go back and wait for Atreyu outside of the swamp?
Beforehand, they didn't know the effect of the swamp.
When Atreyu and Artax first reach the swamp, it looks gloomy and forbidding but no more than that. Despite the name, they didn't realise that anyone entering it would get so depressed that they sank into the swamp.
The little horse uttered one last soft neigh.
"You can't help me, master. It's all over for me. Neither of us knew what we were getting into. Now we know why they are called the Swamps of Sadness. It's the sadness that has made me so heavy. That's why I'm sinking. There's no help."
-- Chapter 3 (C), emphasis mine
Afterwards, it was too late.
By the time they realised what was happening, Artax was already too far gone to be saved. The above quote is from when only the horse's head is protruding from the black water. When they stop, he's already sunk in up to his belly. Most importantly, he doesn't care. Part of the effect of the Swamps of Sadness is depression and apathy, and someone in that state is unlikely to be able to make the effort required to pull themselves all the way out of the bog and back to safety.
"With every step we take, the sadness grows in my heart. I've lost hope, master. And I feel so heavy, so heavy. I can't go on!"
"But we must go on!" cried Atreyu. "Come along, Artax!" He tugged at the bridle, but Artax stood still. He had sunk in up to his belly. And he made no further effort to extricate himself.
"Artax!" cried Atreyu. "You mustn't let yourself go. Come. Pull yourself out or you'll sink."
"Leave me, master," said the little horse. "I can't make it. Go on alone. Don't bother about me. I can't stand the sadness anymore I want to die!"
Artax lacks the willpower even to try to extricate himself, let alone to survive slogging back to the outside of the swamp. Atreyu tries his best to pull him out, but there's nothing he can do. It's not just drowning in the black water that kills Artax; it's his own apathy.