The usual construction for finding torsion elements on complex $K$ theory is using flat vector bundles. So is it still possible to find a simply connected compact space with a nonzero torsion in its $K$ theory. Such an example would be given by a map $f:X\to BU$ which is zero on real cohomology but is not null homotopic. Since the homotopy groups of $BU$ are free abelian, it follows that such a map would be zero on homotopy groups (because $\pi_*(Y)\otimes \mathbb{R}=H_*(Y, \mathbb{R})=H^*(Y,\mathbb{R}$). I don't if such a map could exist or not. I am most interested in the case of a closed simply connected manifold.