Vlad Start page has a wildcard domain at *.proxy.startpage.com.
When users click the anonymous view button, they're taken to a new proxy session URL {id}.proxy.startpage.com
Naturally, websites will have relevative URLs, but since it's a new subdomain, the webserver can automatically proxy all requests as they will hit the proxy server, not the original server.
For non-relative URLs, Start page seems to do some regex edits. Additional protection can be done through a dynamic CSP that only allows connections to the proxy domain.
You won't see this proxy URL when using anonymous view from a first glance because it's encapsulated in an iframe for sandboxing features.