How can I prevent the router from saving my browsing history?

What you mean will not be the browser history, but the name resolution cache, the DNS cache. You can see which websites have been resolved, but not which subpages have been visited, since only the website name is resolved. After a successful resolution, no DNS request is made to the router or its DNS server until the DNS validity expires. The validity is usually several hours, after which the entry is usually deleted when the DNS server does not have other settings like e.g. a longer storage period and/or logging that differs from the validity has activated. In private customer routines, you can usually not view these entries or only view them via the service menu, unless the router brings child protection or the like.

If you want to prevent your router from setting up this buffer, you need another DNS server like e.g. use your own or a public of, for example, Google, Cloudflare, Quad9, Digitalcourage e.V. etc.

DNS requests are unencrypted by default. In order that no one can read it, you need to send your DNS requests encrypted. You can do this either via DNS over TLS (DoT), which can no longer be read, but can easily be detected and blocked, or via the somewhat slower, transparent, DNS over HTTPS (DoH) that cannot be distinguished from website traffic. The latter is supported by today’s large browsers.

This, of course, does not help if the deep package inspection (DPI, deep-packet inspection) is often used with a proxy server, because then encrypted traffic is broken up with specially installed in the certificate memory of the computer and thus familiar root certificates. With these certificates, the error message of the now no longer trustworthy connection is suppressed by the self-issued certificates of the own network inspection certification body. In private home networks, however, this is not done as well, since it is very complicated to set up and operate and even the fewest end customer devices do so.