Apple TV+ has made a number of notable improvements to its web user interface, making browsing content far easier for those using windows PC's, Android devices or versions of MacOS that pre-date Catalina. As discovered by fellow Apple TV aficiando, podcast guest and friend of the site, Benjamin Mayo for 9to5Mac:

As of today’s overhaul, the new TV+ website homepage features the same featured header that you see in the Apple TV app, along with sections for latest releases and genre categories. This makes it feel much more familiar for customers who are used to the native TV app experience.
It’s a good step in the right direction for the service. Nevertheless, tablestakes feature are still missing like the ability to resume from where you left off, see your Up Next queue, or even a search bar. The actual web playback experience and show detail pages are also unchanged as of this update, which is unfortunate. Hopefully, more updates are set to roll out in the coming months.

You can access Apple TV+ through any web-browser by visiting tv.apple.com which requires a credit-card on file for authentication purposes. Whilst we still hold out hope, Apple have made no public mention of any plans to bring the full TV app experience to the web.