Fibrolamellar carcinoma (FLC) is a rare, therapeutically-intractable liver cancer that disproportionately affects youth. Although FLC tumors exhibit a distinct gene expression profile, the chromatin regulatory landscape and the genes most critical for tumor cell survival remain unclear. Here we use chromatin run-on sequencing to discover approximately 7,000 enhancers and 141 enhancer hotspots activated in FLC relative to non-malignant liver. Detailed bioinformatic analyses reveal aberrant ERK/MEK signaling and candidate master transcriptional regulators. We also define the genes most strongly associated with hotspots of FLC enhancer activity, including CA12, SLC16A14,and non-coding RNA LINC00473. Treatment of FLC cell models with inhibitors of CA12 or SLC16A14 independently reduce cell viability and also significantly enhance the effect of MEK inhibitor cobimetinib. These findings highlight molecular targets for drug development as well as drug combination approaches.