I'm trying to search for games themed in the wild west with the tags adult and visual novel. But I can't do that, because there is no way to use filters while doing a search in this site.

Search is title based, not concept or theme based. Try looking for tags that would describe what you look for.

https://itch.io/games/tag-wild-west is only like 80 games, you can look at all of them quickly. Maybe there you will find other tags that connect to the wild west theme.

Also search for wild west only shows about 60 games, I believe you can glance at all those games as well.