Let G be a finite group of Lie type in odd characteristic defined over a field with q elements. We prove that there is an absolute (and explicit) constant c such that, if G is a classical matrix group of dimension n >= 2, then at least c/log(n) of its elements are such that some power is an involution with fixed point subspace of dimension in the interval [n/3, 2n/3). If G is exceptional, or G is classical of small dimension, then, for each conjugacy class C of involutions, we find a very good lower bound for the proportion of elements of G for which some power lies in C.
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