Do most types of organisms evolve at about the same rate?

There is no rate to evolution. Evolution is simply the following of natural selection within an environment. natural selection is like survival of the fittest. A mutation in the organism is caused which can start the natural selection process for that organism. If that organism is able to survive better with that mutation then the trait is passed down and slowly evolution takes place. The only way for evolution to happen at the same rate of another organism is if the organisms rely on each other and one goes through a change. The other dependent species will have to also go through an alteration in order to be able to survive. hope this helps!