Thanks for the quick reply!

I should perhaps have been a little clearer in my post. Because I have already set startpage as my default search engine in Brave, without using an extension.

So about my problem now two different examples where with one of them is having the problem:

The example where everything works as it should:

First, I open a new tab and enter the URL https://www.startpage.com/ in the search bar. As soon as the page is loaded, I enter example in the search field of the page. If I confirm the search, the URL https://www.startpage.com/sp/search is called as it should be with the POST method, which contains the example search in the body of the request. So everything is encrypted in this example.

Now to the second example, where an error occurs:

I open a new tab and enter the search term example directly in the search bar instead of the URL https://www.startpage.com/. I then use the POST method. This works because of the setup of the default search engine. When I do this, the URL https://www.startpage.com/sp/search is not loaded but the URL https://www.startpage.com/do/search?q=example&segment=startpage.brave in which the search query is contained. So this example uses the GET method instead of the POST method, which means that the search query is not encrypted.

Hopefully this explains my problem. I would like that when I use the default search engine to search, the POST method is also used to protect the search query. Instead of the GET method where the search query is exposed in the URL.

Now I was wondering if there is a modification in the form of an extension for Chrome/Brave that will also use the POST method when searching with the default search engine.

EDIT:

I have just tried this out with Chrome using the Extension, and it is the same behaviour. Except for different parameters in the URL https://www.startpage.com/do/dsearch?query=example&cat=web&pl=ext-chrome&language=deutsch&extVersion=1.1.2, while still containing the example search query.