We say a compact Lie group G is simple if it is connected, has finite center and is a simple group modulo its center. We study the relationship between character estimates and the structure of conjugacy classes within G. Suppose G is simple and centerless; the first result shows, for n sufficiently large, the set of n-fold products from a nontrivial conjugacy class contains the identity as an interior point. This n can be chosen uniformly over the set of nontrivial conjugacy classes of G. We use this result to prove a uniform estimate on the set of normalized character values of G. In an opposite direction, we prove a different type of character estimate, which is used to bound the rate of convergence to Haar measure, for certain conjugation-invariant random walks on SU(n). This convergence is with respect to the total variation distance of Diaconis and Shashahani.