I'll write no. My simplest analogy is this...
Imagine your POSTAL MAIL and you encrypting your requests. The mailbox/postman would have to know your encryption and agree to it. But they won't and therefore nothing gets out.
If you want to surf anonymously, then there are anonymizer services that proxy the web content and from the router/ISP/log standpoint, you only went to that web site.
Bob
Hi.
I am on a local network that uses a non-UPnP router and connect to the Internet from there. Is there some way I can block or encrypt my Url requests from being logged by the router machine or the server?

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