Han Solo: Hokey religions and ancient weapons are no match for a good blaster at your side, kid.
Luke Skywalker: You don't believe in the Force, do you?
Han Solo: Kid, I've flown from one side of this galaxy to the other. I've seen a lot of strange stuff, but I've never seen anything to make me believe there's one all-powerful force controlling everything. There's no mystical energy field that controls my destiny.
This exchange takes place in A New Hope, after Obi-Wan Kenobi begins training Luke as a Jedi. He's not the only one who dismisses the power of the Force: Admiral Motti refers to Darth Vader's use of the Force as "sad devotion to that ancient religion."
These statements made perfect sense in the Original Trilogy and the early Expanded Universe, when it seemed that the Jedi had died out long ago. But when the Prequels came out, suddenly the Jedi were in a place of prominence in the galaxy until just 19 years before A New Hope. How could people so quickly forget about the Jedi and their incredible powers?
Numbers of Jedi
While the Jedi are prominent in the Republic and its government, remember that the Star Wars galaxy is a huge place. The population of the capitol of Coruscant alone was up to three million beings at the height of the Republic, and the entire galaxy is home to 100 quadrillion beings.
The exact numbers of the Jedi Order at this time are not specified, but they're small enough that a single Jedi Temple on Coruscant provides enough space to train all the Jedi from shortly after birth until 12 or 13. This includes Jedi younglings that don't make it to Padawan, but get assigned to the Agricultural Corps and other service corps.
Clearly the members of the Jedi Order are an incredibly small percentage of the galaxy's population. Despite their prominent position in the Clone Wars, it's probable that most people in the galaxy, even at the height of the Jedi Order, could live their entire lives without ever seeing a Jedi.
Range of Jedi Influence
The influence of the Republic and the Jedi doesn't extend through the entire galaxy, either. Anakin grows up on Tatooine, for example, a planet run by Hutt gangsters and out of reach of the Republic government. Han Solo grew up on Corellia, which seceded from the Republic at the beginning of the Clone Wars and was largely ignored throughout the war.
While people on planets away from the fighting had no reason to encounter Jedi, people on planets affected by the Clone Wars had a good reason to forget them. To the former Separatists, the Jedi were the villains; to the former Republic, the Jedi were the ones who had turned against the Republic and tried to kill the Chancellor. By the time Han Solo was a young adult, the Jedi had not only disappeared; they had been erased from the popular consciousness.
It's interesting that both Han Solo and Motti refer to the Jedi as a "religion." Even Grand Moff Tarkin, who witnessed Jedi abilities during the Clone Wars, refers to Vader as "all that's left of their religion." So perhaps Han's issue is not with the idea that the Jedi exist, or that they have powers, but merely with their interpretation of the Force.
In many ways, the Jedi do come across as a religion. They have a monastic organization and a strict moral code, and practice meditation to connect with a spiritual being or energy they call the Force.
In the real world, if someone prays for something and it happens, that doesn't prove God exists. In Star Wars, the Jedi can do great things -- but so can beings from numerous other Force traditions that reject Jedi teachings, and so can non-Force sensitives, for that matter. The Jedi view of the Force conflicts with Han Solo's moral code of self-reliance, but he doesn't question the possibility of the Jedi having powers. Based on the place of the Jedi within the Star Wars universe, this seems like a reasonable viewpoint, and probably one that is widely held.