We present Proteus, a novel network-aware approach for optimizing web browsing on heterogeneous multi-core mobile systems. It employs machine learning techniques to predict which of the heterogeneous cores to use to render a given webpage and the operating frequencies of the processors. It achieves this by first learning offline a set of predictive models for a range of typical networking environments. A learnt model is then chosen at runtime to predict the optimal processor configuration, based on the web content, the network status and the optimization goal. We evaluate Proteus by implementing it into the open-source Chromium browser and testing it on two representative ARM big.LITTLE mobile multi-core platforms. We apply Proteus to the top 1,000 popular websites across seven typical network environments. Proteus achieves over 80% of best available performance. It obtains, on average, over 17% (up to 63%), 31% (up to 88%), and 30% (up to 91%) improvement respectively for load time, energy consumption and the energy delay product, when compared to two state-of-the-art approaches.