This paper considers models of conditional moment restrictions that involve non-parametric functions of single-index nuisance parameters. This paper proposes a bootstrap method of constructing confidence sets which has the following three merits. First, the bootstrap is valid even when the single-index estimator follows cube-root asymptotics. Second, the bootstrap method accommodates conditional heteroskedasticity. Third, the bootstrap does not require re-estimation of the single-index component for each bootstrap sample. The method is built on this paper’s general finding that as far as the single-index is a conditioning variable of a conditional expectation, the influence of the estimated single-indices in these models is asymptotically negligible. This finding is shown to have a generic nature through an analysis of Fréchet derivatives of linear functionals of conditional expectations. Some results from Monte Carlo simulations are presented and discussed.