A fairer contest would be to compare cultural evolution with the evolution of organic microbes. A natural experiment is currently doing that on an enormous scale. Hospitals and medical organizations wage constant war on microbes. In some cases, humans care quite a bit about the outcome - it can't be claimed that they aren't trying. This contest gives us some data about the relative rates of evolution in the two realms.
Looking at this data, it's hard to make much of a case for cultural evolution being faster. The extinction rates of disease causing microbes are especially poor. Perhaps, one day, cultural evolution will clearly outstrip organic evolution - but we don't seem to be quite there yet.