Polarization rotator-splitters (PRSs) are crucial components for controlling the polarization states of light in classical and quantum communication systems. We design and experimentally demonstrate a broadband PRS based on the lithium niobate on insulator (LNOI) platform. Both the rotator and splitter sections are based on adiabatically tapered waveguide structures, and the whole device only requires a single etching step. We show efficient PRS operation over an experimentally measured bandwidth of 130 nm at telecom wavelengths, potentially as wide as 500 nm according to simulation prediction, with relatively low polarization crosstalks of $\sim -10\mathrm{dB}$. Our PRS is highly compatible with the design constraints and fabrication processes of common LNOI photonic devices, and it could become an important element in future LNOI photonic integrated circuits.