How did the authorities not find the speakeasy clubs during the prohibition?

With regards to your one pissed off customer scenario, you have to bear in mind these bars were run by the mob. Landlords who decided they did not want to buy from a criminal gang would face violent retribution. An individual customer would be much more easily intimidated. If an individual bar was reported to the police the person complaining, if their identity became known, could have faced trouble from criminals more powerful than the owner of the bar. And this is on top of the fact that the police may not care due to corruption and/or lack of popular support for prohibition described in other answers. The illegal alcohol did not necessarily need to be delivered without anyone seeing it, just no-one who would stick their neck out and report it. Also when raids were conducted, all they would need is a tip off from a corrupt police official and they had the resources to remove all evidence - these groups became extremely well resourced.

1. The law was not universally loved by the Americans. Some stats I've seen showed that as many as 25% of the adult population were drinking illegal alcohol in some form.

2. Several politicians (including perhaps the President of the USA) plus many police liked their occasional drink and the law was an inconvenience to them.

   They were not fanatically motivated to enforce the laws.

3. When there is a ban on a product but strong demand, there is an opportunity for criminal elements to step in. The mafia and other mobs became very powerful during this time.

   Those mobs were generating a lot of cash. So if they had to "invest" some of the profits to payoff the local police chiefs, judges and even mayors, then that was the "cost of doing business".

   What happened in practice was that one or two speakeasies would be shut down temporarily so the mayor and the police could show that they were "tough on crime". The owners were fined and then let go. In a couple weeks, they were back in business in a different location.

Speakeasies weren't just about drinking. Some had gambling and brothels as well. And they had rich and powerful customers who didn't like it if they got arrested too often.

So corruption was indeed an issue, but several other nuances also came into place.

I've always been baffled and fascinated by this "prohibition" era in the USA between 1920 and 1933. That's 13 years!

Let us remind ourselves that we are talking about the 1920s and this is over a century ago; and let us also remind ourselves that the Americas had only been colonised by the West for - what - three centuries by then. Alcoholism was likely to be a problem, particularly in the cities, where it was a dog eat dog world; and thus it was seen as a moral crusade - we see these today, for example, the war on drugs - different intoxicant, but same idea. Wikipedia for example has

Prohibitionists first attempted to end the trade in alcoholic beverages during the 19th century. Led by pietistic Protestants, they aimed to heal what they saw as an ill society beset by alcohol-related problems such as alcoholism, family violence and saloon-based political corruption. Many communities introduced alcohol bans in the late 19th and early 20th century, and enforcement of these new prohibition laws became a topic of debate. Prohibition supporters, called "drys", presented it as a battle for public morals and health. The movement was taken up by social Progressives in the Prohibition, Democratic, and Republican parties and gained a national grassroots base through the Woman's Christian Temperance Union. After 1900, it was coordinated by the Anti-Saloon League.

The New York Times ran an article in 1989 with the headline 'Prohibition was a success' and the encyclopedia, Alcohol and Temperance in Modern History indicated that alcohol consumption declined substantially due to Prohibition and that rates of liver cirrhosis, alcoholic psychosis, and infant mortality also declined.

How did the authorities somehow not find these? ... Unless the authorities were actually "in on it"

Given that drinking is the social activity of choice in the West it would not be surprising that sometimes the authorities turned a blind eye to some such activities; after all, not every drinking den is a den of iniquity - judgement always has to be exercised; and no doubt, there is the odd corrupt cop who was in the pay of the mafia ...

Prohibition was enacted largely at the behest of pressure groups. This was a very new political tactic in the US where a group uses an emotional issue to create themselves a sizable cadre of single-issue voters, who they can deploy surgically to unseat any politician who isn't in a completely safe district who opposes them.

Arguably this tactic had been tested out with reconstruction racial politics, but it was honed to a science by Wayne Wheeler and his Anti-Saloon League (ASL). A politician's voters might mostly be against prohibition, but if they aren't willing to vote a politician out over it, and the ASL cruises in with 10% of the electorate who are willing to vote him out if he doesn't support it, the choices for most were either kowtow to the ASL, or be replaced by another politician who would do so.

What I'm leading up to here is that Prohibition was imposed on the USA, and particularly the cities, via these new pressure groups. This was in the days before scientific polling, so we don't know for sure, but its quite likely it never had a majority of support in the country. We do know the "wet" politicians who resisted the ASL were largely clustered in the cities.

So the reason why enforcement was so lax was that by and large city authorities didn't support prohibition. Furthermore, their voters didn't support it. The ASL actually had to push through whole new Federal law enforcement agency responsibilities just so they could have an enforcement agency that actually cared about enforcing the laws they had pressured the Federal Government into passing.