The first part of the study will be dose escalation of emavusertib in combination with standard of care (SOC) cisplatin, gemcitabine, and durvalumab. Once the expansion dose of emavusertib is determined, the expansion part of the study will open. All patients will be treated with emavusertib in combination with SOC cisplatin, gemcitabine, and durvalumab for up to 8 total cycles. Patients may receive no more than 8 total cycles of gemcitabine and cisplatin for BTC. Up to 2 cycles of gemcitabine and cisplatin, with or without durvalumab, may be given off protocol prior to enrollment; therefore, some patients may only receive 6 or 7 cycles of gemcitabine, cisplatin, and durvalumab in combination with emavusertib on study. After 8 cycles, patients will discontinue cisplatin and gemcitabine and continue maintenance emavusertib and durvalumab.