irules to rewrite old url to new url without changing the users browser url

Reformatting your iRule for discussion:

when HTTP_REQUEST { 
    if { [HTTP::uri] starts_with "/test-site" } { 
        HTTP::header replace "Host" "newsite.test.com" 
        HTTP::path "/alternatesite/test" 
        pool newsite-pool-HTTP 
    } else { 
        pool oldsite-pool-HTTP 
    } 
}

What you're basically saying is if the URI starts with /test-site, change the Host header, change the HTTP path, and send to a different pool. Otherwise send to the oldsite pool. I would start by changing HTTP::path to HTTP::uri. The HTTP::path is the part of the URI before any query strings, while HTTP::uri is the entire uri. The above iRule will work for the simplest environments, as long as the new site only ever uses the /alternatesite/test URI. If it doesn't, and it's in any way dynamic, then you have to do some more work. Depending on the complexity, you may want to take a look at the ProxyPass iRule (v11.3 and below) or the rewrite profiles in 11.4 and above.

You may also want to consider enabling a OneConnect profile.