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Tourism is a complex dynamic system including multiple actors which are related each other composing an evolving social network. This paper presents a growing bipartite network model that explains the rise of the supply network in a tourism destination from the beginning phases of development. The nodes are the lodgings and services in a destination and a link between them appears if a representative tourist hosted in the lodging visits/consumes the service during his/her stay. The specific link between both categories are determined by a random and preferential attachment rule. The analytic results show that the long-term degree distribution of services follows a shifted power-law distribution. The numerical simulations show slight disagreements with the theoretical results in the case of the one-mode degree distribution of services, due to the low order of convergence to zero of X-motifs. The model predictions are compared with real data coming from a popular tourist destination in Gran Canaria, Spain, showing a good agreement between numerical and empirical data.