Let's say some xyz website's server was just another Signal client. I can talk to it from my Signal client in a way that is not subject to State surveillance. But I can't do that using HTTPS as my security model.
As to how Signal does it... Read their blog.
What is insecure about a browser is that you can't be sure who is at the other end of the communication. That includes man-in-the-middle attacks, which are basically a variant of the case where you are not talking to the person you think you are.
That problem with Signal is solved by proving the identity of the person you are talking to out-of-band with the application, for example by linking up with your friend while you are standing next to each other, or verifying the link by a phone call.
I hope you can see that there's no directly analogous way to secure a web session for the consumer web, since the process of verifying the identity of the server manually would never scale. The solution to this problem -- ta da -- is to create authorities who you trust who will independently certify the server's identity -- certificate authorities, if you will. And yes, once you add CA's in the middle, the line of trust becomes an n-tree of trust, and you have a much higher risk of things going wrong.
If you want a fully secure Web experience -- delete all CA's except any that you independently audit. Then manually validate the SSL certs of every server you want to talk to and add it manually to your cert store. Now you have a Signal-like experience for the web. Good luck.