(No this is not a thread of the questions of authorship, its clearly old Bill from Stratford who penned them).
It has become somewhat common to diss or defend popular entertainments of less than steller quality with the term “well its not Shakespeare”. But, as I recall, Shakespeare was a writer of plays meant for mass appeal, as can be gleaned from the fact that every second line seems to be a contemporary euphemism or a pun for/on copulation, defection or genitalia.
What would have been the high brow entertainment in Shakespeare’s time? Stuff by Marlow or Jonson?
The university playwrights looked down on Will. He didn’t respect the classical Three Unities, he didn’t understand that gore and violence were to be kept off-stage, and simply reported on by a herald or other actor, and so on. He was writing for the masses, not for the learned people who could truly appreciate Drama!
Whether that is a fair summary or not is a matter of dispute. There was the group of « University Wits » who were educated at the universities and then drifted to professional drama in London, which some see as a transition to Shakespeare ‘s approach.
I think it is fair to say Shakespeare’s career overlapped a time of change in how popular theater was viewed. The fact that officials banned theaters from within London’s city walls in 1572, and in 1575 any such performances, is often cited as a good example of antipathy toward the medium. This is a little bit erroneous–the 1572 ban was actually a public health measure taken to try and reduce the risk of plague spreading, but there is truth to the idea it was seen as a “low” form of entertainment. Theaters often shared common areas of towns with brothels and bear pits, which were certainly low brow affairs.
On the flipside by the end of Shakespeare’s life high nobility and even royalty were attending plays fairly commonly, and by 1600 Shakespeare was receiving contemporary praise from academic authors and others who would have been part of the upper crust of society. The various publishing collections of Shakespeare’s works in the first hundred or so years after his lifetime were often sponsored or dedicated to noble patrons.
So…it was seen as low brow but it was gaining popularity in polite society and was getting formal praise and recognition too, and it was in a state of such transition for all of Shakespeare’s life. There was a later reactionary movement against theater when the Puritans took over in the 1640s, as well.
The “high brow” drama of Shakespeare’s life was “court theater”, which was inspired by Greek and Roman classics, and also by French theater, it was largely acted and performed by university students who were all well educated in the classics and had formal educations, and most people who attended university at this time were part of society’s upper classes.
Shakespeare’s contemporaries and he himself were part of what is often now called “Elizabethan Theater”, and was popular theater for the masses, typically produced and run by troupes of professional actors who formed companies. The origins of these groups were in the itinerant traveling acting troupes which had been popular for a couple hundred years prior, and that often historically had put on really low brow theater for common rooms of inns, small village market fairs etc.
Based on the reaction by some with lives in the “court theater”, positively, to Shakespeare, it must be the case that at least by 1600 there was widespread respect for the most talented playwrights of the popular stage, even among the more rarefied air of the court theater types.
What we need is a Fretful Porpentine! 
[Moderating]
Moving from FQ to CS.
Genuinely high-brow stuff was written in Greek or Latin, as it had been for the last two thousand years. It had been only a few hundred years since Dante had had to put a lot of work into trying to prove that poetry in Italian was OK. Heck, he had had to put a lot of work into proving that poetry in Italian was poetry at all. (And when he did it, he wrote in Latin, so he’d be taken seriously.)
Ben Jonson thought he was top tier literature, for sure. Read his poem written after Shakey died.
In the documentary Shakespeare in Love Marlowe is revealed to be the master dramatist of the time. He even offered Shakespeare the plot for Romeo and Ethel the Pirate’s Daughter, which Shakespeare later adapted.
Whereas in the competing documentary Upstart Crow, we learn that Shakespeare actually wrote most of the plays attributed to Marlowe, who was too busy womanizing and hunting for Catholic spies to actually do any writing.
High-brow stuff like Asinaria ? 
From the viewpoint of Elizabethan/Jacobean intellectuals? Yes. Plautus was in the syllabus, after all. The Western Canon of Shakespeare’s day.
More like Sidney and Spenser. At the beginning of Shakespeare’s career, if you had an ambition to be regarded as a “serious” writer, you wrote poetry – and young Shakespeare did do that, publishing a couple of short epics and dedicating them to an aristocratic patron, before he decided to devote himself to writing for the stage full time. At the time, that definitely meant choosing the popular-but-ephemeral over the “literary.” (The literary gatekeepers of the day recognized that drama could have literary merit, but they generally didn’t think the stuff being performed in their own time, before popular audiences, did. In Sidney’s Defense of Poesy, written a decade or less before Shakespeare began his career, he offers a ringing defense of the literary and social functions of comedy and tragedy, but all of his positive examples are from the classical Greek and Roman world; contemporary comedy is dismissed as something that “naughty play-makers and stage-keepers have justly made odious.”)
But by the end of Shakespeare’s career, people were at least starting to consider contemporary, English plays from the public stage to be “serious” literature. If they hadn’t, we wouldn’t have about half of Shakespeare’s plays at all: they were printed for the first time in the First Folio, which was essentially a fancy coffee-table book, with dedicatory poems from other writers and an introduction by its editors that urges the buyer to “read him … and again, and again.” It is not an acting edition, and would have been much too large and unwieldy to be useful for that purpose; the assumption, by 1623, was that people would be interested in sitting down to read Shakespeare’s plays, and that he was a poet / playwright of some significance. The most famous of those dedicatory poems, Ben Jonson’s “To the Memory of My Beloved the Author, Mr. William Shakespeare” compares him favorably to a whole range of writers, including classical playwrights like Aeschylus and Sophocles, widely-admired English poets like Chaucer and Spenser, AND other English playwrights (Marlowe, Kyd, Beaumont) – with the implication that all of these people represent the best in literary style.
To be sure, Jonson had a distinctly pro-drama-as-serious-literature agenda – he had already published his own plays alongside his poetry in similar fancy-coffee-table-book style as The Works of Benjamin Jonson, a decision that raised some eyebrows among his contemporaries, one of whom commented that Jonson clearly didn’t know the difference between works and plays. And he got mad at audiences who preferred plays that he considered unworthy, including some of Shakespeare’s later works, so he didn’t think everything Shakespeare wrote was gold. However, he wasn’t the only one of Shakespeare’s contemporaries to praise Shakespeare by comparing him to the classical Greeks and Roman dramatists, or to suggest that his plays had lasting value as literature. About midway through Shakespeare’s career, Spenser’s friend Gabriel Harvey noted that “The younger sort takes much delight in Shakespeare’s Venus, & Adonis: but his Lucrece, & his tragedy of Hamlet, Prince of Denmark, have it in them, to please the wiser sort"; Harvey, in other words, is dismissing one of the poems as fluff for younger audiences and suggesting that a play exceeds it in intellectual and literary merit and is roughly the equivalent of Shakespeare’s other long poem, a fairly weighty work based on Roman history.
At around the same time as Harvey, though, you did get stuff like The Return from Parnassus, a play by one of the “university playwrights” that @Northern_Piper references, which pokes fun at Shakespeare’s popularity and burlesques his fellow actors as unlettered ignoramuses who admire him because they don’t know any better. (Presumably, though, the Cambridge-educated author’s point was that HE was writing the good stuff – a play that reflected his own classical learning and superior wit rather than catering to the popular taste – which implies a certain baseline agreement that English drama could do that. And also, the fact that he satirizes people who thought Shakespeare was The Bestest Playwright Ever implies that this was a recognizable attitude.)
So, it’s complicated! Probably the best way to sum it up is that Shakespeare, and his fellow popular playwrights, were working in an emerging genre that they knew people took less seriously than classical literature, contemporary English poetry, or even English drama that was written by educated writers for private performance. But, through some combination of self-promoting ambition and sheer talent, they managed to make their audiences take it seriously, all the same. It’s not clear how personally invested Shakespeare was in having people regard his plays as “important” literature; other than those two dedications to his early poems, what we have is what other people wrote about him and his works, not what he said himself. But by the end of his lifespan, people had in fact started to think of him that way.
As corroborated by even more documentary footage…
“'Tis not I who penned this dreadful play. 'Twas Christopher Marlowe!”
“Thou already tried to use that line of argument when the Jews wanted to kick yo’ ass after The Merchant of Venice!”
I remember reading spéculation that The Tempest, one of his last plays, was sort of a « sod off » to the classicists: « See, I know perfectly well about the Three Unities and çan write a play that respects them, when I want to. I just don’t need to. »
Shakespeare’s plays are so well liked in part because they have something for everyone. Not every other line is ribald, but most plays have both easy humour and complex symbolism. Shakespeare doubtless attracted a lot of professional jealousy, borrowed from other plays liberally, wrote in English but coined new phrases and words constantly, had a remarkable understanding of society and human nature neatly summarized in verse, yet was so successful at entertaining it still lives on unchanged in its envisioned form after half a millennium. Since few things do that, few things are Shakespeare