We consider the problem of estimating the conditional density $\pi$ of a response vector $Y$ given the predictor $X$ (which is assumed to be a continuous variable). We provide an adaptive nonparametric strategy to estimate $\pi$, based on model selection. We start with a collection of finite dimensional product spaces, spanned by orthonormal bases. But instead of expanding directly the target function $\pi$ on these bases, we prefer to consider the expansion of $h(x,y)=\pi(F_X^{-1}(x),y)$, where $F_X$ is the cumulative distribution function of the variable $X$. This 'warping' of the bases allows us to propose a family of projection estimators easier to compute than estimators resulting from the minimization of a regression-type contrast. The data-driven selection of the best estimator $\hat{h}$ for the function $h$, is done with a model selection device in the spirit of Goldenshluger and Lepski (2011). The resulting estimator is $\hat{\pi}(x,y)=\hat{h}(\hat{F}(x),y)$ otherwise, where $\hat{F}$ is the empirical distribution function. We prove that it realises a global squared-bias/variance compromise, in a context of anisotropic function classes: we establish non-asymptotic mean-squared integrated risk bounds and also provide risk convergence rates. Simulation experiments illustrate the method.