"The Sands of Time", a novella by P. Schuyler Miller, first published in Astounding Stories, April 1937, available at the Internet Archive. Any of these covers look familiar?
A sequel, "The Coils of Time", was published in Astounding Science-Fiction, May 1939, also available at the Internet Archive.
Many decades ago, I read a short story about a time traveler who asks an archaeologist for help. He convinces the archaeologist that he can time travel by bringing him back a freshly killed dinosaur (small one.)
After failing to convince the paleontologist (who narrates the story) with photographs and dinosaur eggs, he makes another trip and brings back an archaic bird:
Some of the cocksure exuberance had gone out of him. He was thinner, and his face was covered with a stubbly growth of beard. He wore shorts and a tattered shirt, and his left arm was strapped to his side with bands of some gleaming metallic cloth. Dangling from the fingers of his good hand was the strangest bird I had ever seen.
He flung it down at my feet. It was purplish-black with a naked red head and wattled neck. Its tail was feathered as a sumac is leafed, with stubby feathers sprouting in pairs from a naked, ratty shaft. Its wings had little three-fingered hands at the joints. And its head was long and narrow, like a lizard's snout, with great, round, lidless eyes and a mouthful of tiny yellow teeth.
I looked from the bird to him. There was no smile on his lips now. He was staring at the footprints of the rock.
"So you've found the beach." His voice was a weary monotone. "It was a sort of sandy spit, between the marshes and the sea, where they came to feed and be fed on. Dog eat dog. Sometimes they would blunder into the quicksands and flounder and bleat until they drowned. You see—I was there. That bird was there—alive when those dead, crumbling bones were alive—not only in the same geological age but in the same year, the same month—the same day! You've got proof now—proof that you can't talk away! Examine it. Cut it up. Do anything you want with it. But by the powers, this time you've got to believe me! This time you've got to help!"
I stopped and picked the thing up by its long, scaly legs. No bird like it had lived, or could have lived, on this planet for millions of years. I thought of those thirty photographs of the incredible—of the eggs he had had, one with an embryo that might, conceivably, have been an unknown genus of turtle.
The time traveler needs help because when traveling back in time to the age of the dinosaurs he met a girl and wants to transport her back to the present. However only one person can travel in time at a time so he needs the archaeologist at the other end to wait for the girl.
Every road has its ending. Noon found them standing beside the leaden hulk of the Egg, face to face with reality. One of them, and only one could make the journey back. The Egg would not hold two, nor was there power enough in its accumulators to carry more than one back through the barrier between the time coils. If the girl were to go, whe would find herself alone in a world unutterably remote from her own, friendless and unable to understand or to make herself understood. If Donovan returned, he must leave her here alone in the Cretaceous jungle, with no food, no means of protection from man or beast, and no knowledge of what might be happening sixty million years later which would seal her fate for good.
There was only one answer. Her hand went to his arm and pushed him gently toward the open door of the Egg. He, and he alone, could get the help which they must have and return to find her. In six hours at the outside the Egg should be ready to make its return trip. In that six hours Donovan could find me, or some other friend, and enlist my aid.
[. . . .]
"I'm leaving in ten minutes," he said. "The batteries are charged."
"What can I do? I'm no mechanic—no physicist."
"I'll send her back in the Egg," he told me. ""I'll show you how to charge it—it's perfectly simple—and when it's ready you will send it back, empty, for me. If there is any delay, make her comfortable until I come."
The archaeologist agrees to help. Unfortunately after the traveler goes back he and the girl find themselves in the middle of a battle between aliens on Earth.
The trouble with aliens happened on the previous trip:
He did a lot of thinking on that journey through blackness. He put two and two together and got five or six different answers. Some of them hung together to make sense out of nightmare.
First, the girl herself. The rocket, and Donovan's faith in a science that he was proving fallible, told him that she must have come from another planet. Her unusual strength might mean that she was from some larger planet, or even some star. At any rate, she was human and she was somebody of importance.
Donovan mulled over that for a while. Two races, from the same or different planets, were thirsting for each other's blood. It might be politics that egged them on, or it might be racial trouble or religion. Nothing else would account for the fury with which they were exterminating each other. The girl had apparently taken refuge with her bodyguard on this empty planet. Possession of her was important. She might be a deposed queen or princess—and the blacks were on her trail. They found her and laid siege—whereupon Terry Donovan of 1937 A. D. came barging into the picture.
[. . . .]
It was the sky Donovan feared now. Dinosaurs they could outwit or outrun. He thought he could even fight one of the little ones, with her to cheer him on. But heat rays shot at them from the sky, with no cover within miles, was something else again. Strangely enough, the girl seemed to be enjoying herself. Her voice was a joy to hear, even if it didn't make sense, and Donovan thought that he got the drift of her comments on some of the ungainly monstrosities that blemished the Cretaceous landscape.
But we don't find out what happened after the last trip back, with the narrator waiting for him:
I noted carefully everything he did, every setting of every piece of apparatus, just as he showed them to me. Then, just four hours after he threw that incredible bird down at my feet, I watched the leaden door of the Egg swing shut. The hum of the generators rose to an ugly whine. A black veil seemed to envelop the huge machine—a network of emptiness which ran together and coalesced into a hole into which I gazed for interminable distances. Then it was gone. The room was empty. I touched the switch that stopped the generators.
The Egg did not return—not on that day, nor the next, nor ever while I waited there. Finally, I came away. I have told his story—my story before—but they laugh as I did. Only there is one thing that no one knows.
They are chased by a dinosaur and never make it back to the present. The archaeologist meanwhile is digging for evidence that the traveler went back in time and the story or my recollection of it ends with that.
Two lines of footprints come down across the wet sands of that Cretaceous beach, side by side. Together they cross the forty-foot slab of sandstone which I have uncovered, and vanish where the rising tide has filled them. They are prints of a small, queerly made sandal and a rubber-soled hiking boot—of a man and a girl.
A third line of tracks crosses the Cretaceous sands and overlies those others—huge, splayed, three-toed, like the prints of some gigantic bird. Sixty million years ago, mighty Tyrannosaurus and his smaller cousins made such tracks. The print of one great paw covers both the girl's footprints as she stands for a moment, motionless, beside the man. They, too, vanish at the water's edge.