There was a supreme court decision that said that the government can't conspire with private companies to censor. Can anyone tell me the name of the case?
I'm not convinced that there is a U.S. Supreme Court case with such a clear holding. For example:
May a private entity running a public access channel ban speakers based on the content of their speech—something a government entity running the same channels could not do?
Yes, the Supreme Court held in a 5-4 opinion in Manhattan Community Access Corporation v. Halleck (2019).
Why? Because the First Amendment doesn’t apply to private entities in this instance.
You may be looking for one of the cases discussed in Halleck which notes (in the official syllabus) that:
"A private entity may qualify as a state actor . . . when the entity exercises “powers traditionally exclusively reserved to the State.” Jackson v. Metropolitan Edison Co., 419 U. S. 345, 352 . The Court has stressed that “very few” functions fall into that category. Flagg Bros., Inc. v. Brooks, 436 U. S. 149, 158 .
Per the official syllabus in the link above, Jackson held that:
The convergence between the actions of a state and a heavily regulated private utility company do not rise to the level of state action if the utility company has a partial monopoly in providing electrical service and uses a procedure that the state utilities commission finds to be appropriate under state law in terminating service to customers.
As the linked material explains:
A private entity performing a public function can be classified as a state actor, but merely being regulated and overseen by the state does not equate to performing a public function if the company is managed by private actors. In this decision, the Court placed limits on the public function doctrine, which otherwise could have grown into a massive exception permitting the inference of state action.
Thus, heavy state regulation of a private entity still doesn't constitute state action which can violate a constitutional right, in this case, even though if the state has had greater control over the nominally private entity (e.g. appointing its board of directors), the private entity's actions would be state action.
Per the official syllabus in the link above, the primary holding of Flagg Bros. was that:
A warehouseman's proposed sale of goods entrusted to him for storage, as permitted by [New York State's Uniform Commercial Code] § 7-210, is not "state action," and since the allegations of the complaint failed to establish that any violation of respondents' Fourteenth Amendment rights was committed by either the storage company or the State of New York, the District Court properly concluded that no claim for relief was stated by respondents under 42 U.S.C. § 1983.
In reaching this conclusion, the U.S. Supreme Court reasoned that:
The challenged statute does not delegate to the storage company an exclusive prerogative of the sovereign. Other remedies for the settlement of disputes between debtors and creditors (which is not traditionally a public function) remain available to the parties. Though respondents contend that the State authorized and encouraged the storage company's action by enacting [the statute], a State's mere acquiescence in a private action does not convert such action into that of the State.
In other words, enactment of a state statute authorizing private action is not state action that can violate a constitutional right.
I figured it out myself. It was Bantam Books, Inc. v. Sullivan, 372 U.S. 58 (1963)