The big problem with Godot 4.0 in web (HTML5) is that the latest Godot just support the Vulkan render-backend, but most of modern browsers dont work with Vulkan.
Most of web and mobile applications are just build on top of OpenGL and OpenGL ES but currently there is no solution to render Vulkan-Shaders reliable and stable in browsers.
The lastest nightly-builds of some browsers (Firefox, Chrome) are able to use "WebGPU". With this implementaion it will be possible to build modern games for Browsers... but this feature is far away from a stable and usefull state.
Regards
ProggerParrot