The first intuition as to how to increase your anonymity in privacy would be to use a similar model as email; just like an "anonymizing proxy" it would seem effective to use a web proxy that could strip out your identifying information like your IP address, cookies, your web browser version, what web page you last visited, etc. As an added bonus, the proxy could cache recently retrieved documents and thus even possibly speed up your web browsing.This is, in fact, the case.
However, there are not very many publicly available free anonymizing proxies. If one did become popular, it would probably choke very quickly from having to retrieve thousands of web pages a second for users and send the pages back to tens of thousands of users a second. While such a system might be convenient, it would require cash inflow to maintain; the only reasonable way to do this would likely be to insert advertisements into the web pages it proxied, but the legal and ethical issue of forcibly inserting banner advertising into someone else's content is rather questionable. Consequently, it seems unlikely that a free, high-quality web proxy will be forthcoming.