The claim that individuals believe themselves to be better drivers than average is one of the common pieces of evidence used to show the effect of overconfidence bias. The BPS Research Digest reports that the experimental method of these results may be slightly flawed.
Instead of asking students to compare themselves to the overall group (or average), they asked the students to compare somebody else to the group. By this manipulation, they found that students still showed a preferential bias towards the individual over the group whether they were that individual or somebody else was.
This result doesn't imply that individuals are not overconfident--we still are--but merely that the cause of that overconfidence may not be narcissism but instead a failure to compare an individual to a fair population mean.