We investigated flows of visitors migrating between different portal subpages. Two various portals were studied as weighted networks where nodes are portal subpages and edge weights are numbers of user transitions. Such networks differ from networks of portal subpages connected by hyperlinks prepared by portal designers. Distributions of link weights, node strengths, and times spent by visitors at one subpage follow power laws over several decades for data collected during two different days and for weekly data. The distribution of numbers P(z) of unique subpages visited during one session is exponential and there is a square-root dependence between the total number of transitions n during a single visit and the average z . A model of portal surfing is developed where the browsing process corresponds to a self-attracting walk on the weighted network with a short memory. Results of numerical simulation are in agreement with weekly and daily portal data, and our analytical approach fits empirical data in the center part of scaling regime.