There are enough extant plays by Shakespeare's contemporaries that we can make a decent comparison of their respective styles and subject matter. The Elizabethan and Jacobean dramatic tradition was a little different from modern theater in the way play-texts were disseminated -- there wasn't a single master manuscript written by Shakespeare from which quarto and folio texts were simply transcribed, or at least not one that's survived. Print editions had to be compiled at the time they were prepared for publication (perhaps from memory, perhaps from rough drafts or scripts retained by actors) and this accounts for some of the discrepancies in the exact wording of the text of any given play. But this wasn't so far back in history that we don't have a significant corpus of other Elizabethan/Jacobean drama, or that we can't compare Shakespeare's works to other popular fiction and nonfiction of the time. 400 years is a long time, but it's not so long that Shakespeare's works are all that remains as a tool to understand the literary landscape of that time.

We have more of Shakespeare's works as surviving texts than we have works by some of his contemporaries, but is that because of his prolific authorship, his popularity in his own time, or some other quirk? Shakespeare was more prolific than some of his contemporaries, less prolific than others. Some of Shakespeare's own plays are considered lost, like Cardenio, co-written with John Fletcher. It's not impossible that there are others, especially plays where Shakespeare was only one of several collaborators. Shakespeare and his contemporaries weren't so precious about their works that they made detailed lists of what they worked on and when; most of the work of ferreting out what got written when and by who has fallen to scholars of Early Modern drama. We also have no grave shortage of other works by Shakespeare's contemporaries. Ben Jonson wrote 20 plays, perhaps more, as well as dozens of masques and plenty of poetry; it's hard to say how many plays Christopher Marlowe might have written had he lived longer, but enough of his work survives to justify plenty of scholarship and discussion of its internal themes and stylistic qualities. Thomas Middleton might easily have written as many plays as Shakespeare, many of them still extant today, and John Fletcher was likewise a prolific writer and prolific collaborator. This is just skimming off the top; there are loads more playwrights whose work in this period is still extant, either in whole or in part, and no shortage of lost plays either. (How plays get lost during Shakespeare's time is a different question, and one for somebody other than me.) To an ardent student of Early Modern drama, that's still not enough to get the fullest possible picture, but it's not nothing; Shakespeare's plays are not the sole remaining remnant of a larger literary tradition. If you remove all plays indisputably written by Shakespeare or predominantly by Shakespeare from the total catalogue of all mid-16th and early 17th century English drama, there's still a lot to go on -- some of it good, some of it great, some of it bad, and some of it mediocre. That's a pretty normal distribution for art being made at any time, and I don't think there's reason to be overly concerned that it's a self-selecting pool of works considered Great Works. It's more of a testament to reading broadly --- you have to branch out if you want to get a picture of what Shakespeare and his contemporaries held in common as well as how they differed.

Maybe it's easier for work to get published broadly if people are really clamoring for it, and highly desirable works of drama during this period might even get bootlegged -- the various surviving editions of Shakespeare's plays does tell us something about which of his plays were popular with certain audiences, and that to some extent they were profitable ventures overall. People wanted to see his plays performed and they also wanted to read them. Was Shakespeare a better writer than his contemporaries? In his own day he was a more bankable writer than some of them, and perhaps a less bankable writer than others -- he was far from the only working playwright, and the same companies and impresarios that were paying money for his plays were also paying a host of other writers for their work. His work was highly regarded enough to have high-profile fans, and to be performed for state occasions before high-profile visitors to the English court. In his own time (or at least relatively soon after his death) some of his contemporaries were comfortable criticizing Shakespeare as an author; Ben Jonson supposedly described him as "wanting art", lacking restraint and failing to exercise an attention to factual detail -- still other 17th century writers criticized him for his violation of Classical dramatic unities by piling on subplots and cramming the historical events of decades into a three-hour running time. Shakespeare wasn't beyond reproach in his own time and afterward, and much of the school of thought that builds Shakespeare up to be the ne plus ultra of Early Modern drama and in fact all theater ever is a later invention of the 18th and 19th centuries. That view of Shakespeare's works turns into a hazy kind of historical revisionism really fast -- there were many other authors besides Shakespeare working during Shakespeare's lifetime and people liked those other authors' works enough to pay to watch them performed. Shakespeare didn't own the game, and there were authors writing challenging and powerful plays in styles very different from Shakespeare's, as well as authors writing semi-predictable comedies and tragedies.

I like Shakespeare a lot, but I think the popular focus on Shakespeare as the best writer in the English language ever to exist, or the best Early Modern dramatist, distorts the picture. People can be so focused on Shakespeare as the best Early Modern dramatist that other Early Modern English writers get treated like sideshows, or discussed solely in terms of how they compare to Shakespeare. Our expectations regarding Elizabethan drama have been shaped by Shakespeare, but had Shakespeare never existed there would still have been Christopher Marlowe and Ben Jonson -- and then John Webster, John Ford, Thomas Kyd, George Chapman, and many more. Plays by Shakespeare's contemporaries are still being studied and performed all over the world, and at their best they're really damn good. Whether these works are cumulatively or individually better than Shakespeare's works is a subjective literary judgment, but Shakespeare wasn't the sole great writer rising out of a pile of dross, he was one of many competitive, collaborative individuals working in a particular timeframe.

There's always room for further answers, but there's also been some great answers about Shakespeare on /r/AskHistorians which should help provide context for Shakespeare's acclaim. Firstly, /u/cdesmoulins discusses Shakespeare's competition for 'best playwright of the era', /u/sunagainstgold explains how we got from 'Shakespeare's just some playwright' to 'Shakespeare's a genius', while /u/texpeare discusses how perceptions of Shakespeare have changed over the centuries. Additionally there's plenty of really interesting writing on other aspects of the historical Shakespeare in our FAQ.

Isn’t there a lot of evidence that ‘Shakespeare’ might have ‘been’ his contemporaries, in that many famous writers were involved in the creation of Shakespeare’s anthology?