Werewolves Do More Damage on the Full Moon
Werewolves obviously don't want people to know their true identities, so they hide away during the nights where they are partially transformed. I'd assume they only have ears or a tail during the times where there is a crescent moon, so all they'd need is a decent hat or something, and they'd easily be able to hide that away. It gets more difficult the more the moon is revealed, but they'd probably learn to deal with it after a while, and families of werewolves would probably spend their entire lives figuring out how to keep it concealed.
The full moon is different. They completely lose their minds, and all logic goes out the window. Nothing can restrain them during this time, so this is the only time when people get to see the real transformation.
This is probably the only time where people see the death and destruction these werewolves can cause. If they've completely lost all control during a full moon, then it stands to reason that they'd also cause a lot of death and destruction in their wake.
This is what people remember every time they think of the werewolves. They think of the death, loss, and trauma that they faced when the wolves last went on a rampage.
This leads to the common myth that werewolves only exist during the full moon because that is the only time non-werewolves even see them. Every other day of the year the werewolves are trying not to be spotted. Probably because they are ashamed of what they have done, and horrified at their own natures.
Having 1 of 20 people being a wolf actually makes things easier for them to hide, because there are more people on their side trying to make sure they are not being caught. I'd imagine whole organizations existing to help werewolves avoid capture.
The werewolf organizations might even purposefully perpetuate the myth. People might not fear them as much if they think wolves are only dangerous one day out of the month.
Obviously, people who hate and fear the wolves are never going to make any attempt to understand them. In the minds of everyone else, they are probably nothing more than mindless monsters, so they would easily accept and believe any number of untrue myths about them.
People believe patently untrue things all the time, and it's not like a real werewolf would ever step in and confirm or deny these things. Then they'd probably be killed for sure.
Townsfolk who hate and despise the wolves would come up with all manner of rumors and lies to make people hate them even more, and none of the poor wolves would ever be able to do anything about it.
"Did you know a werewolf can be killed by a silver bullet? You'd better buy my whole supply of silver bullets."
"Did you know werewolves only come out during the full moon? Daylight is when they're at their weakest, so you have to kill them then."
"Did you know werewolves are 20 feet long, spit acid, and love the taste of human flesh? Even in their human forms, they can't get enough of the flavor, so they cook and eat children."
While there may be truth to some of the claims, people are blinded by their own biases. They'll believe whatever people tell them to believe. If popular opinion says wolves only come out at the full moon, that is the truth they will accept.
