Network models of language have provided a way of linking cognitive processes to the structure and connectivity of language. However, one shortcoming of current approaches is focusing on only one type of linguistic relationship at a time, missing the complex multi-relational nature of language. In this work, we overcome this limitation by modelling the mental lexicon of English-speaking toddlers as a multiplex lexical network, i.e. a multi-layered network where N=529 words/nodes are connected according to four types of relationships: (i) free associations, (ii) feature sharing, (iii) co-occurrence, and (iv) phonological similarity. We provide analysis of the topology of the resulting multiplex and then proceed to evaluate single layers as well as the full multiplex structure on their ability to predict empirically observed age of acquisition data of English speaking toddlers. We find that the emerging multiplex network topology is an important proxy of the cognitive processes of acquisition, capable of capturing emergent lexicon structure. In fact, we show that the multiplex topology is fundamentally more powerful than individual layers in predicting the ordering with which words are acquired. Furthermore, multiplex analysis allows for a quantification of distinct phases of lexical acquisition in early learners: while initially all the multiplex layers contribute to word learning, after about month 23 free associations take the lead in driving word acquisition.