the King's Men acting company. He wrote 37 plays for them, several with collaborator john Fletcher. Often referred to as being obscure in his own day, he was actually quite popular with the playgoing public. Although his plays belonged to the company, and not to himself, people would make choices of which playhouse to frequent (and there were several in Southwark, London, along with non-theatrical entertainments, such as bear-baiting and brothels, with which the playhouses had to compete) based on whether Shakespeare had written that day's presentation.
Nothing is better for proving popularity in one's own day than the fact that Shakepeare also made enemies, at least one of whom published a scathing paper denouncing Shakespeare as an upstart, and warning people off his plays.
If the whole vast thing was a cover up, some proof of it would have surfaced in the centuries since. Instead conspiracy theorists come up with strange, weird, arcane, and usually incomprehensible 'clues' that someone else was the author.
The sonnets and other poems were recognized in their own day as being by Shakespeare. And reputable scholars of his work are in agreement that the author of the poems and the author of the plays are one and the same.
The members of The King's Men company paid a great deal of money to print his plays as a memorial to him several years after he died. If The First Folio had denied credit to the 'true' author, public notice would have been paid not long after the book hit the stands. Instead, the conspiracy theory that Shakespeare was actually someone else didn't start until more than a century after he died.