Could Lex Luthor and The Joker best the Avengers?

Definitely.

At least short term. By being evil villains, neither can or will have permanent besting power over any of the good people.

A supervillain with no known definite physical superpowers like Lex Luthor or the Joker has to be pretty bad apple to take down A superhero team like the Justice League.

I think they could best the Avengers. Hulk is a powerhouse character. I don't think they'd be overly concerned about dealing with the Hulk who is part of the team.

Luthor had Solomon Grundy on his team, the Legion of Doom, and Luthor knows the weaknesses of his teammates. Grundy is basically the Hulk's DC counterpart. If he knows how to control Grundy he can control the Hulk.

Joker is smart enough to know how to control the Hulk. There was some classic DC/Marvel crossover in which the Joker pitted the Hulk against Batman if I am not mistaken.

Luthor and Joker know how to exploit opponents' weaknesses.

Precisely, which is why they can only succeed short term. Most superhero good people, have a tendency to take a while to figuring out and then crafting a solution to conquering the enemy. It makes for good story telling and edge of the seat suspense. (usually)

I don't recall ever seeing or reading about The Joker defeating the Justice League. At best, in some continuities, he succeeds in corrupting a hero like Superman (as seen in the game Injustice), but it those wins usually leads to his death (in Injustice, Superman kills the Joker).

Currently in the comics, Lex Luthor is a Martian/Human hybrid in order to become an Apex predator. In this form, Luthor has proven to be a powerful villain against the League.

Luthor, in his traditional form, and The Joker would have difficulty beating the Avengers. Luthor's key trait, his high level intellect, is rivaled by several of the Avengers (Iron Man, Ant-Man, Black Panther). The Avengers have experience fighting evil geniuses and corrupt businessmen. Luthor would be easy to fight.

The Joker is unpredictable, but he is still powerless. Again, most of the Avengers have fought against mentally insane villains before, so The Joker is not much of a threat.

Other than strength, Grundy is a completely different character from the Hulk. The only DC character that is similar to the Hulk is the antihero Damage, who can turn into a monster for an hour.
https://dc.fandom.com/wiki/Ethan_Avery_(Prime_Earth)

Okay. Has Luthor ever bested the Justice League?

Not physically and never alone.

He usually hides behind legal loopholes and publicity, so that the League can't touch him. It was how he was elected US President in the comics and was almost elected in the Justice League animated series.

Even in the recent comics, Luthor is not fighting the League alone; he reformed The Legion of Doom and acquired several artifacts that would allow him to rule the world.

Besides, Luthor cares about only defeating Superman because he believes humanity is not reaching to its fullest potential due to relying on a powerful alien. The only other heroes Luthor fought against are Batman, Martian Manhunter, and maybe Wonder Woman.

Batman Wants To Join The Marvel Universe.

So Luthor has only worn a supersuit when fighting Superman?

Has Joker ever used a supersuit?

Joker has never used a super suit, as far as I know.

He mainly deals with Batman and his vigilante allies, so he has no use of a supersuit.

Luthor only uses his armor when dealing with Superman and other Kryptonians like Supergirl.

I know in DCAU cartoons Joker has defeated the Justice League at least once.

Not sure about Luthor.

Which episode? I only recall one episode where Joker confronted the League, and that was the Las Vegas bomb episode. He also didn't fight them except for Batman; he kept them busy with the time bombs and used the Royal Flush Gang to fight the League.

Luthor, powered by Brainiac, did fight the League in the second season finale of Justice League Unlimited, but he was defeated by The Flash

I don't recall exactly- it was from Justice League or Justice League Unlimited.

I've seen both.

He only had major appearances in two Justice League episodes: "Injustice For All" (where he helped capture Batman for the Injustice Gang) and "Wild Cards" (where he placed bombs all over Las Vegas).

He had no appearances at all in Justice League Unlimited.

https://dcau.fandom.com/wiki/Joker/Chronological_appearances

Okay. What's the episode I am thinking of from Justice League in which Joker beat the JL and the JL was sent to space in a rocketship set to explode?

I have no clue. You may be confusing several episodes together.

This was back in the early 2000's. I don't remember the episode in detail.

Are you talking about the Superman Animated Series episode where Joker and Luthor fought Superman and Batman?

At the end of that crossover, Joker stole an experimental plane in the shape of a smile.

No.