We propose a general method to derive Berry-Esseen bounds of the correct order in limit theorems for random sums of independent and centered random variables that have a finite absolute moment of order \(2+\delta \) for some \(\delta \in (0,1]\) . Our technique combines generalizations of the classical Berry-Esseen theorem with new estimates on the Kolmogorov distance between Gaussian random sums and the limiting scale mixture of the standard normal distribution. This method is illustrated by means of three more concrete situations, including the approximation of a geometric sum of centered random variables by a Laplace distribution. For \(\delta =1\) , the resulting Berry-Esseen bound turns out to be of the optimal order \(p^{1/2}\) in the i.i.d. case.