Current bandwidth selectors for kernel density estimation that are asymptotically optimal often prove less promising under more moderate sample sizes. The point of this paper is to derive a class of bandwidth selectors that attain optimal root-n convergence while still showing good results under small and moderate sample sizes. This is achieved by minimizing bias corrected smoothed bootstrap estimates of the mean integrated squared error. The degree of bias correction determines the rate of relative convergence of the resulting bandwidth. The bias correction targets finite sample bias rather than asymptotically leading terms, resulting in substantial improvements under smaller sample sizes. In a simulation study, the new methods are compared to several of the currently most popular bandwidth selectors, including plug-in, cross-validation and other bootstrap rules. Practical implementation, and the choice of the oversmoothing bandwidth are discussed.