What is a word for a punishment given by people without proper trial instead of the court?

Is there a word that describes the giving of punishment (usually stoned to death) of a criminal by the local people near where the crime happened?

A typical example would be when a thief in a village steals a motorcycle, and then gets caught by the angry mobs, and then usually the thief gets either killed by the mobs or beaten badly without proper hearing and trial, usually until the police intervene.

I've tried searching for "trial by the mass" (but apparently Google found mass trial instead), "public trial" (but apparently it means something different), "trial without hearing" (but the results are not what I'm looking for), "punishment by the mass" (but Google found capital punishment and collective punishment instead), and so far, nothing gives me a word for punishment by the public/mass.

From Merriam-Webster Online:

Lynch

transitive verb : to put to death (as by hanging) by mob action without legal sanction

From Online Etymology:

lynch (v.)

1835, from earlier Lynch law (1811), likely named after William Lynch (1742-1820) of Pittsylvania, Virginia, who c.1780 led a vigilance committee to keep order there during the Revolution. Other sources trace the name to Charles Lynch (1736-1796) a Virginia magistrate who fined and imprisoned Tories in his district c.1782, but the connection to him is less likely. Originally any sort of summary justice, especially by flogging; narrowing of focus to "extralegal execution by hanging" is 20c. Lynch mob is attested from 1838. The surname is perhaps from Irish Loingseach "sailor." Compare earlier Lydford law, from a place in Dartmoor, England, "where was held a Stannaries Court of summary jurisdiction" [Weekley], hence:

Lydford law: is to hang men first, and indite them afterwards. [Thomas Blount, "Glossographia," 1656]

(Which I find interesting—I'd always assumed lynch was related to the actual noose somehow.)

Google Ngram purports to find a reference to "lynch law" going back to 1790, but that is suspicious. The terms really start to heat up about 1830.

And an article I was just reading in Scientific American reminded me that murder is also a perfectly appropriate term for the taking of life extralegally.

A typical example would be when a thief in a village stole a motorcycle, and then get caught by the angry mobs, and then usually the thief got either killed by the mobs or beaten so badly without proper hearing and trial...

I think you might be looking for the term vigilante justice:

someone who takes the law into his/her own hands by trying and/or punishing another person without any legal authority. In the 1800s groups of vigilantes dispensed "frontier justice" [also called vigilante justice or street justice] by holding trials of accused horse-thieves, rustlers and shooters, and then promptly hanging the accused if "convicted."

Vigilante justice has been glamorized since before "Robin Hood", the earliest edition having been printed

between 1492 and 1534, but shows every sign of having been put together from several already existing tales. (Wikipedia)

Not a single word but a useful idiom: Take the law into your own hands :

• to do something illegal in order to punish someone.

  • Her mother took the law into her own hands when she heard that her child had been abused. She decided to take the law into her own hands and rescue the dog from its owner, who beat it.

• Usage notes: usually said about someone who does something because they believe that the authorities will not take action

( Cambridge Dictionary of American Idioms)

extrajudicial is the legal term for this sort of behaviour

adjective

• (of a sentence) not legally authorised.
• (of a settlement, statement, or confession) not made in a court of law.

The word for punishment meted out outside the formal and legal justice system is extrajudicial.

It's often used to describe a state actor which chooses to perform an act outside the legal system in punishing its opponents.

"The prince of Florin took matters into his own hands to perform the extrajudicial killing of his rival in love."