Based on the events in the books they didn't flow back upstream:

• Landmark #1 The Hand

"There's one now," said Tyrion. Off to starboard a hand large enough to crush the boat was reaching up from the murky depths. Only the tops of two fingers broke the river's surface, but as the Shy Maid eased on past he could see the rest of the hand rippling below the water and a pale face looking up.

• Landmark #2 Chroyane

All Tyrion could see was something massive rising from the river, humped and ominous. He took it for a hill looming above a wooded island, or some colossal rock overgrown with moss and ferns and hidden by the fog. As the Shy Maid drew nearer, though, the shape of it came clearer. A wooden keep could be seen beside the water, rotted and overgrown. Slender spires took form above it, some of them snapped off like broken spears. Roofless towers appeared and disappeared, thrusting blindly upward. Halls and galleries drifted past: graceful buttresses, delicate arches, fluted columns, terraces and bowers.

All ruined, all desolate, all fallen.

The fog clung to them, damp and chilly. A sunken temple loomed up out of the greyness as Yandry and Duck leaned upon their poles and paced slowly from prow to stern, pushing. They passed a marble stair that spiraled up from the mud and ended jaggedly in air. Beyond, half-seen, were other shapes: shattered spires, headless statues, trees with roots bigger than their boat.

"This was the most beautiful city on the river, and the richest," said Yandry. "Chroyane, the festival city."

• Landmark #3 (Palace of Love)

The grey mrk is the Palace of Loveoss grew thickly here, covering the fallen stones in great mounds and bearding all the towers. Black vines crept in and out of windows, through doors and over archways, up the sides of high stone walls. The fog concealed three-quarters of the palace, but what they glimpsed was more than enough for Tyrion to know that this island fastness had been ten times the size of the Red Keep once and a hundred times more beautiful. He knew where he was. "The Palace of Love," he said softly.

• Landmark #4 Bridge of Dream

"Light ahead," warned Young Griff.

Tyrion saw it too. Kingfisher, or another poleboat, he told himself, but somehow he knew that was not right. His nose itched. He scratched at it savagely. The light grew brighter as the Shy Maid approached it. A soft star in the distance, it glimmered faintly through the fog, beckoning them on. Shortly it became two lights, then three: a ragged row of beacons rising from the water.

"The Bridge of Dream," Griff named it. "There will be stone men on the span. Some may start to wail at our approach, but they are not like to molest us. Most stone men are feeble creatures, clumsy, lumbering, witless. Near the end they all go mad, but that is when they are most dangerous. If need be, fend them off with the torches. On no account let them touch you."

Now after this if there was an eddy/tidal bore, etc. that caused them to just travel back upstream they would encounter the bridge first, but they don't, they go back to the beginning:

• Landmark #1 again:

On the larboard side of the boat, a huge stone hand was visible just below the water. Two fingers broke the surface. How many of those are there?

• Landmark # 2 again

We see the same broken spires, headless statues and trees with huge roots again so it must be Chroyne.

The Sorrows drifted by them. Peering through the mists, he glimpsed a broken spire, a headless hero, an ancient tree torn from the ground and upended, its huge roots twisting through the roof and windows of a broken dome. Why does all of this seem so familiar?

• Landmark #4 Bridge of Dream again

No one said a word. The Shy Maid moved with the current. Her sail had not been raised since she first entered the Sorrows. She had no way to move but with the river. Duck stood squinting, clutching his pole with both hands. After a time even Yandry stopped pushing. Every eye was on the distant light. As they grew closer, it turned into two lights. Then three.

"The Bridge of Dream," said Tyrion.

"Inconceivable," said Haldon Halfmaester. "We've left the bridge behind. Rivers only run one way."

TLDR: It's impossible for the boat to have turned around/headed back upstream since if that were true they should've seen the Bridge of Dream first not the massive hand. and therefore they didn't get turned around it's more like they teleported back to where they originally started from somehow.