When I switch language, the content is changing, but the menu is not changing from NL to ENG and vise versa.
Can I somehow force to update the menu as well? I tried disabling AJAX for the complete website, which resolves the menu issue, but then parts of the website are broken...
Usually, this kind of issue is resolved by simply using the Store a language cookie to support language filtering for AJAX option from Dashboard → WPML → Languages → Language filtering for AJAX operations. Can you tell me if anything changes while using this option, please?
As I can see, the theme that you're using (Anneliese) is not a part of the WPML compatible themes list (https://wpml.org/documentation/theme-compatibility/). This means that some compatibility issues might still be between this and WPML.
I also checked the description of the theme and I could see that it doesn't mention anything about compatibility with multi-language plugins or even translation. In this case, you can try to reach out to the theme authors and ask them to apply for the Go Global program here https://wpml.org/documentation/theme-compatibility/go-global-program/. This way, they will work with our team to make sure that everything works fine between the theme and WPML.
I'm sorry but WPML cannot change the way that the theme handles the menu. This is a part of the website that's created by the theme you are using and changing how this is working can only be changed from the theme side.
In this case, I think that the only option is to use the theme option for disabling AJAX. If the theme author is responding back and asks for any change from our side, we are here to help. As I mentioned in my previous message, our compatibility team is ready to work with the theme authors to make sure that everything works very well between WPML and the theme that you're using.