I am looking to build applications on the web with Unity's WebGL. Unfortunately it is very heavy and not at all praiseworthy for the past couple of years. Instead I am learning three.js and Babylon.js as they provide lightweight web applications. My feedback is that Unity should work on their WebGL department since most of the users are awaiting for it. Please give us an update as to when we will be seeing Unity's new WebGL.
They are hiring persons for a new web application project "Project Tiny". But I believe that we have to wait more than one year.
Indeed classic Unity could be considered as the "big gun" for web exports. Three.js and Babylon.js are currently positioned to be more lightweight, although they do not then carry the same featureset that Unity has. It is unfortunately a world of tradeoffs overall. Project Tiny that @AgnosiaGames mentions is a new development from Unity that aims to complement Unity's offering for a more lightweight web-oriented deployment target. Give it a go if you can see it evolve in the direction that you'd be interested in!
You can also take a look at "PlayCanvas" @saifshk17 I feel like it's a middle-ground between Unity and Three.js/Babylon.js. Cheers