Rollup merge of #81379 - GuillaumeGomez:improve-urls, r=Nemo157

Rollup merge of #81379 - GuillaumeGomez:improve-urls, r=Nemo157

Improve URLs handling

Fixes #81330.

Explanations: before this PR, when emptying the search input, we still had `?search=` in the URL, which wasn't very nice. Now, if the search is empty, we drop the `?search=` part.

Also, I realized while working on this PR that when we clicked on a menu link when we were on the search results, the search parameters would look like: `?search=#the-anchor`, which was super weird. Now, it looks like this: `?search=the-search#the-anchor`.

Also, I didn't use the `Url` very nice API because it's not available in any IE version (sadness...).

cc `````@lzutao`````
r? `````@Nemo157`````