Studies of language evolution in the lab have used the iterated learning paradigm to show how linguistic structure emerges through cultural transmission--repeated cycles of learning and use across generations of speakers . However, agent-based simulations suggest that prior biases crucially impact the outcome of cultural transmission. Here, we explored this notion through an iterated learning study of English-French bilingual adults (mostly sequential bilinguals dominant in English). Each participant learned two unstructured artificial languages in a counterbalanced fashion, one resembling English, another resembling French at the phono-orthographic level. The output of each participant was passed down to the next participant, forming diffusion chains of 10 generations per language. We hypothesized that artificial languages would become easier to learn and exhibit greater structure when they were aligned with participants' bilingual experience (i.e., English languages being easier to learn overall), or as a function of practice (i.e., languages learned second being easier to learn overall). Instead, we found that English-like languages became more structured over generations, but only when they were learned first. In contrast, French-like languages became more structured regardless of the order of learning, suggesting the presence of an asymmetric switch cost during artificial language learning. Moreover, individual differences in language usage modulated the amount of structure produced by the participants. Overall, these data suggest that bilingual experience impacts how novel languages are learned at an individual level, which can then scale up to cultural transmission of novel language at a group level.