The problem is not evolution per se, but the mechanism behind it. It would take BILLIONS, not mere millions, of years for the complexity of the human genome to have evolved based upon random mutation and subsequent selection. The problem is one of irreducible complexity. You can't select a more complex organism from a simpler one unless each successive generation of mutation has been selected as providing a competitive advantage. In an irreducibly complex organism, removal of any constituent part renders the whole useless. These organisms are extremely hard to explain using evolutionary theory. If mutations are random, that takes far too long to account for the diversity of life on earth and the complexity of the human genome. Intelligent design is not the same thing as creationism, it is just more consistent with some of creationism's ultimate presumptions. Most adherents of ID tend to accept the theory of common descent (anathema to creationists). What ID proponents point to are flaws in evolutionary theory that cannot account for irreducibly complex organisms, where many, many mutations had to have occurred to produce a complicated physical manifestation, yet one cannot simply remove one aspect and still have functionality. As a simple analogy, think of a pencil evolving into a pen (I know, I know, I'm just trying to explain). From one generation to another, mutations in the pencil would have to occur which would provide some benefit to the pencil for that mutation to be "selected" by nature as more fit. But having a spring, or ink, or thumb-press would have no use except in the final product (ink would need the tube, for example). Many organisms show these kind of traits, traits that could not have evolved by the result of random mutation, or that under the most generous of scenarios would take a hundred times longer to have evolved than evolutionary theory provides. The alternative, according to ID, is that the process is the product of design. Every attempt by evolutionists to claim they've "debunked" the irreducible-complexity argument that I've read is loaded with flaws and tends to prove the argument more than disprove it, on careful reading, despite the authors' stated conclusions.
Straw man. Most genome changes are not the result of nor rely on random mutation, though you would be correct that random mutation is a pretty damn slow. Some of the other mechanisms provide smooth and rapid DNA drift, which almost certainly dominates evolutionary variation in most complex organisms.