Currently (September 2023), Apple’s iTunes store seems to have a bug where it’s impossible to subscribe to new podcasts.

However, there is a workaround.

Find your desired podcast on the iTunes store.

Click the “get” button, on the right side of the episode list, for any episode of the podcast.

Switch to your library.

The episode should show up, along with an entry for the podcast itself.

Right-click on the podcast title, and select “Subscribe Podcast”.

iTunes will not fetch the list of all available episodes, as usual.