(Spoilers All) Why are so many people convinced Dragon-riders are a thing?

Is there any text to support the dragon-rider concept?

Yes, it is not at all a baseless concept.

Though young to the throne, Jaehaerys revealed himself from an early age to be a true king. He was a fine warrior, skilled with lance and bow, and a gifted horseman. He was a dragonrider as well, riding upon Vermithor—a great beast of bronze and tan who was the largest of the living dragons after Balerion and Vhagar. Decisive in thought and deed, Jaehaerys was wise beyond his years, always seeking the most peaceable ends.

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The singers dubbed her the Realm's Delight, for she was bright and precocious—a beautiful child who was already a dragonrider at the age of seven as she flew on the back of her she-dragon Syrax, named for one of the old gods of Valyria.

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Among the greens, it was said that they were obviously the sons of Breakbones, and many doubted whether they could be dragonriders.

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Among these new dragonriders was Addam of Hull—a brave and noble youth who was brought by his mother, Marilda of Hull, to try for a dragon along with his brother Alyn.

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But of all the new dragonriders, the worst were the drunkard named Ulf the Sot, who took the name Ulf the White once knighted,

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For while Aemond had been marching on Harrenhal, Daemon had met Queen Rhaenyra and her dragonriders over King's Landing, their dragons wheeling above the city.

Aegon and his sisterwives Rhaenys and Visenya are perhaps the most famous of all dragonriders; riding dragons is how they were used in battle.

Taming and the subsequent mastery and riding of dragons is how they became machines of war. You can't reason with a dragon, you can't get your point across to it, you can't tell it 'go burn those nasty men' or something. You ride it, bend it to your will, and it will do the rest.

/r/asoiaf Thread