We perform an empirical study of the preferential attachment phenomenon in temporal networks and show that on the Web, networks follow a nonlinear preferential attachment model in which the exponent depends on the type of network considered. The classical preferential attachment model for networks by Barabási and Albert (1999) assumes a linear relationship between the number of neighbors of a node in a network and the probability of attachment. Although this assumption is widely made in Web Science and related fields, the underlying linearity is rarely measured. To fill this gap, this paper performs an empirical longitudinal (time-based) study on forty-seven diverse Web network datasets from seven network categories and including directed, undirected and bipartite networks. We show that contrary to the usual assumption, preferential attachment is nonlinear in the networks under consideration. Furthermore, we observe that the deviation from linearity is dependent on the type of network, giving sublinear attachment in certain types of networks, and superlinear attachment in others. Thus, we introduce the preferential attachment exponent $β$ as a novel numerical network measure that can be used to discriminate different types of networks. We propose explanations for the behavior of that network measure, based on the mechanisms that underly the growth of the network in question.