I said he no clue that Qui-Gon was a Jedi, not that he hadn't heard of them. All that time Watto was dealing with a Jedi and he had no idea who Qui-Gon was. For that matter, Anakin only figured it out after he saw Qui-Gon's lightsaber.
Here's something to think about: Han Solo grew up under the Imperial regime, one that he knew lied as a matter of course. And to many people, the difference between the Republic and the Empire may have been one of names and labels. It's possible that Han did wonder if the Jedi were real -- not because he hadn't heard about them, but because he was wondering if the government of the Republic was as dishonest and unreliable as the government of the Empire. Maybe he was thinking in terms of, "Yeah, the Republic claimed it had an order of warrior-monks able to do fantastical things with the Force, but I don't see any evidence the Force is real. I think it was the Republic's propaganda, just as untrustworthy as anything from the Emperor's mouth."
In AOTC, Mace Windu told Yoda he would "take what Jedi we have left" to Geonosis, which to me implied that there were already barely any active Jedi remaining at the start of the Clone Wars. I suppose maybe the intent was that he was referring to just the Jedi they had on hand on Coruscant, with the majority spread out across the galaxy on various missions and such.
Mace took those that were at the temple to Geonosis. 200 Jedi. Of which about 20 lived. The rest of these 10,000 Jedi were scatted across the Galaxy and those younglings pressed into service early as padawans like Ahsoka Tano and Kanan. The preteen and teenaged commanders of the Grand Army of the Republic under their Jedi Knight generals.
That was ten years before order 66, therefore 30 years before Han Said what Han said. Both Republics may be a lot smaller than we thought. The Empire might be a lot smaller than we thought. What if there are massive civilized areas of occupied space adjacent to the Empire/Republic where the Jedi are bared or excluded? Example: Green Lanterns are banned from Earth's Solar System in the 31st century. No idea why. Example: After the Monks of the Foundation assimilated the four Kingdoms of the Periphery by pushing atomic science with strings, the nearby ramshackle worlds descended into barbarism, didn't feel like saying "Yes" to enslavement, as they started digging for coal to keep their cities warm. Consider what the Jedi did, and why it's a bad idea to let them run freely through your space. They listen to the voices in their head and kill without thinking whenever some aspect of the universe disturbs their right and wrong. They're not police. They don't enforce law written down in a book. They just whimsically cut limbs off anyone who seems like they might be guilty of doing something. And they're allowed to do this? Are they like Bond? A license to kill (and maim). Also... I can only see that the Jedi will "look after" worlds that pay the Jedi to do so in children. Quid pro quo. Qua Gon Gin said the Republic screenings would have identified Anikan as a force user when he was much younger, but think about that. Every child is tested, and any child that is found to be a potential Jedi is taken. On one hand maybe the parents are paid, as their children are dragged away never to be seen again, on the other hand their children are taken, and most parents really, really don't like that. Although untrained potential Jedi arn't much to worry about, unless a Sith gets hold of them? Are the screenings to recruit Jedi or starve out the Sith?
I sort of doubt it's a "Storm-Trooper-esque" "we're taking this child to make him work and fight for us" sort of thing (as we're told about in EpVII.) I think it's more like, "your child has special abilities so we're letting him into our school for gifted individuals." I wouldn't think they'd then admit the kids and then never let them contact their families again; I'd think the Jedi would see how that'd push someone's emotions in the wrong direction. Hell, even when Qui-Gon sets his eyes on Anakin he tries to get both him and his mother, but is unable to. This suggests that Anakin would have had some contact with his mother while he was training to be a Jedi. As for why he never visited his mother once training? I dunno, Tatooine is so far away and remote, he got busy doing things, you know he wanted too.... but things are rough this time of year. I'll come see you next year, mom, I promise!
The force is actually nutty shit, it just happens to be true. Even if he did witness force powers first hand, he dismissed them as magic tricks. There wasn't a massive overthrow of the planet 20 years ago, and you didn't spend the last 20 years scrounging for a living while schools and universities are used as cartel strongholds & the news media broadcasts messages from the ministry of truth while anyone who questions any of it is executed on the spot. So the modern 1st world is probably not the BEST example. I could bring in a few specific real world examples even in the 1st world, but I get the feeling that bringing up specific real-world politics is probably not a good idea! Han probably grew up in the outer rim - beyond the reach of the Old Republic anyway. It would be like a Chinese peasant having a perfect understanding of American politics immediately following WWII.
I believe that is relevant to the discussion : 5 Ways Growing Up in North Korea Is Crazier Than You Think
Everybody in the galactic senate knows about them, as they were sent by the senate to Naboo. The Trade Federation knew about them, as they invited them aboard their ship to "negotiate". The Trade Federation droid identified them as Jedi based on a single meeting. Anakin knew Qui-Gon was a Jedi because of his light saber. So a presumably uneducated eight year-old slave on an outer rim world knows about the Jedi, and what weapons they use. Anakin was confident enough that the average citizen on Coruscant knows who the Jedi are that he announced to a bar full of people that he was on "Jedi business". This after a high-speed chase where Anakin used his light saber to cut into someone's car roof (no doubt caught on hundreds of surveillance cameras). Dozens of Jedi have a huge light-saber battle in the middle of an arena within view of hundreds of bug creatures, who no doubt went home to tell their bug family and friends about what happened. So I think there is sufficient evidence from the prequels that the Jedi were very well known at the time. I'm not saying that they couldn't go from the Star Wars version of ambassador/Navy Seal/super heroes to the Star Wars version of The Freemasons in a few decades. I'm just saying it is extremely unlikely. Such as whom?
They were certainly well known on certain planets and known to higher government officials. We do have to remember that this series spans an entire galaxy. That's a lot of distance to cover, and everybody's exposure to certain things is going to be wildly different. So I'd wager that the Jedi were both well known and not known at all depending on where you are and who you ask.
Again, Anakin an 8-year-old slave on an outer-rim world knows about Jedi and the weapons they carry. If there's anyone we can point at as an example of someone who should have no knowledge of the Jedi it's him. Even Watto, his "blue-collar", junk-yard-owning master on an outer-rim world knows of Jedi and the "mind-tricks" they play and dismisses Qui-Gon as being one. It seems at least suggested to me in PT that Jedi are known about and accepted in the galaxy as peace-keepers or and inter-stellar force that goes around helping out and doing various things. Less than 20 years later, they're a myth. It seems hard to believe they'd be seen this way after such a short time. Han grew up partially during the time of the Jedi and any adults he was around would have known about them and heard about them. You can suppress media, reading materials, news, etc. but you can't suppress people's memories and ability to tell stories about the Jedi and how they were totally around doing things and the role they played in the collapse of The Republic.
It's not like Han was completely oblivious to the Jedi. He just didn't believe the stories until he personally took part in one and could see it for himself.
How much does the average American citizen know about the Navy SEALs? What if tomorrow, the American government disbanded the SEALs, executed or jailed every current and former SEAL, wiped all mention of them from public record, sealed all their records, took down their website, branded them "enemy combatants" and criminalized any mention of them on social media? What would our grandkids know about Navy SEALs? Not much. They might even think they were just a myth. If we could conceivably imagine that happening here in the US, imagine how easy it would be for a massive, draconian government to cause the memory of entire Order to erased, to the point that hot-shot young hipster smugglers doubt that the Jedi even exist.
^Or celebrities. How much do you think kids today know about the Beatles? Humans are not exactly known for their long attention spans.