Large language model (LLM)-based conversational agents (CAs), with their advanced generative capabilities and human-like conversational interfaces, can serve as reading partners for children during dialogic reading and have shown promise in enhancing children's comprehension and conversational skills. However, there is limited research on the efficacy of LLM-based bilingual CAs in children's language acquisition in English as a Foreign Language (EFL) contexts. This randomized controlled trial study investigated the effectiveness of LLM-powered CAs compared with traditional parent--child shared reading in promoting engagement and improving learning outcomes among children with EFL. An interactive e-book featuring a LLM-powered CA was developed to engage children in dialogic reading through questioning and scaffolding. Sixty-seven children, aged 5 to 8, were randomly assigned to either an experimental (AI-led) group or a control (parent-led) group. The study found that children in the experimental group outperformed the control group in reading comprehension, with comparable benefits in vocabulary acquisition and story retelling, both immediately and in delayed tests. In the meantime, this study unpacks children's different engagement patterns when reading with the CA versus reading with their parents. Children reading with the CA demonstrated higher behavioural engagement and visual attention, while those in the parent-led group showed greater affective engagement and narrative-relevant vocalizations. The findings highlighted insights into the potential of LLM-powered CAs in children's language acquisition and suggested key design implications for developing better CAs for children from multilingual backgrounds.