The modeling of extreme events arises in many fields such as finance, insurance or environmental science. A recurrent statistical problem is then the estimation of extreme quantiles associated with a random variable $Y$ recorded simultaneously with a multidimensional covariate x in R^d, the goal being to describe how tail characteristics such as extreme quantiles or small exceedance probabilities of the response variable Y may depend on the explanatory variable x. Here, we focus on the challenging situation where Y given x is heavy-tailed. Without additional assumptions on the pair (Y,x), the estimation of extreme conditional quantiles is addressed using semi-parametric method. More specifically, we assume that the response variable and the deterministic covariate are linked by a location-dispersion regression model Y=a(x)+b(x)Z where Z is a heavy-tailed random variable. This model is flexible since (i) no parametric assumptions are made on a(.), b(.) and Z, (ii) it allows for heteroscedasticity via the function b(.). Moreover, another feature of this model is that Y inherits its tail behaviour from Z which thus does not depend on the covariate x. We propose to take profit of this important property to decouple the estimation of the nonparametric and extreme structures. First, nonparametric estimators of the regression function a(.) and the dispersion function b(.) are introduced. This permits, in a second step, to derive an estimator of the conditional extreme-value index computed on the residuals. A plug-in estimator of extreme conditional quantiles is then built using these two preliminary steps. We show that the resulting semi-parametric estimator is asymptotically Gaussian and may benefit from the same rate of convergence as in the unconditional situation. Its finite sample properties are illustrated both on simulated and real tsunami data.