We derive generalization error bounds for the training of two-layer neural networks without assuming boundedness of the loss function, using Wasserstein distance estimates on the discrepancy between a probability distribution and its associated empirical measure, together with moment bounds for the associated stochastic gradient method. In the case of independent test data, we obtain a dimension-free rate of order $O(n^{-1/2} )$ on the $n$-sample generalization error, whereas without independence assumption, we derive a bound of order $O(n^{-1 / ( d_{\rm in}+d_{\rm out} )} )$, where $d_{\rm in}$, $d_{\rm out}$ denote input and output dimensions. Our bounds and their coefficients can be explicitly computed prior to the training of the model, and are confirmed by numerical simulations.