Speakeasies were everywhere. They were ignored by the authorities, at least in major cities, 99% of the time. When they got raided, they moved down the street, changed their name, and reopened. What’s remarkable about Prohibition is how totally openly people violated the law. People who wouldn’t go into saloons in the past now started frequenting them. Commentary at the time constantly railed at the law-breaking and predicted doom. (Just like every other decade since.)
Night clubs also violated the law virtually every day. Sometimes they did it without subterfuge, other times they poured alcohol into cups when you ordered tea, or they allowed, as dolphinboy said, people to bring in their own bottles (probably actually flasks). The times called for entrepreneurial cleverness and owners responded with flair and originality.
The twenties added an additional new factor. Women mostly did not go to bars before Prohibition; night clubs and speaks were new ways for middle-class couples to go out together for live entertainment. Their proliferation was less a replacement of what came earlier than a modern adaptation to changing conditions.