Is there a website out there where I can find/compare data on stellar parameters for the brightest stars? I'm looking for some kind of catalog/repository which lists stellar parameters (e.g. mass, radius, temperature) in a similar vein to NASA's Exoplanet Catalog (but except for stars), for the brightest stars in the sky, with $m_v&lt;4$. I'd rather not have to refer to Wikipedia as they only list data from one source (or an average of several uncited sources).
I'm aware of SIMBAD, but that site doesn't have a very intuitive UI and doesn't list mass measurements. I've also tried CDS (http://cdsportal.u-strasbg.fr/), but I can't find distance measurements from Gaia (although apparently Gaia doesn't measure distance and I need to put some kind of external study into a Gaia search query which I haven't got a clue how to use).
As you’ve discovered, there are a lot of different catalogs out there, and many ways to search them.  It would help if you could narrow your question a bit - what kind of data are you looking for?   You seem to want stellar masses at least - what else?  As noted above, there are few stars for which mass is measured directly, but there are robust ways of estimating it from other information, just as effective temperature can be determined from spectral type or photometry.
One catalog that might meet your needs is the TESS Input Catalog (TIC) or its brighter subset, the TESS Candidate Target List.  It compiles information from a lot of other sources, and includes magnitudes from optical through infrared, as well as estimates of stellar mass, radius, temperature, metallicity, luminosity, and surface gravity.  It also includes stellar parallax (i.e., distance) and proper motion from Gaia.
In addition to the ways of searching given in the above link, you can search it through Vizier here - I find the Vizier interface a little easier to navigate than MAST, though both have a lot of options.
Now this catalog may be overkill for what you want: it has 471 million stars, though “only” 27 million of those have mass estimates, and “only” 2 million have distances.    :-)  So knowing more about what you want could help narrow things down.
Edit: The above numbers refer to an earlier release of the catalog, prior to Gaia DR2.  The updated version (which is what I linked to from Vizier) has about 1.7 billion stars, of which 1.27 billion have distances and 455 million have mass estimates.
Edit 2:  After clarification in the comments that the goal is finding information only for very bright stars, this isn’t the best catalog to use.  While it does include entries for some bright stars (e.g. Vega, Betelgeuse), the information included is very limited, presumably because they are saturated (overexposed) in most surveys.
In general it is much harder to get homogeneous data for very bright stars because of this.  There is a wealth of data on them individually, but because they are saturated in systematic surveys, there aren’t uniform sources of data on them to which known relations can be applied in bulk.
The Yale Bright Star Catalogue (searchable here) is a good source of basic data (photometry, spectral type, distance, multiplicity), but for derived quantities like mass or radius you would have to apply known relationships yourself (if you need the data in bulk) or go to individual references for particular stars if you don’t need a lot of them. Honestly Wikipedia isn’t a bad starting point in the latter case - it is generally clear about what references the numbers come from.