Disorder is emerging as a strategy for fabricating random laser sources with very promising materials, such as perovskites, for which standard laser cavities are not effective or too expensive. We need, however, different fabrication protocols and technologies for reducing the laser threshold and controlling its emission. Here, we demonstrate an effectively solvent-engineered method for high-quality perovskite thin films on a flexible polyimide substrate. The fractal perovskite thin films exhibit excellent optical properties at room temperature and easily achieve lasing action without any laser cavity above room temperature with a low pumping threshold. The lasing action is also observed in curved perovskite thin films on flexible substrates. The lasing threshold can be further reduced by increasing the local curvature, which modifies the scattering strengths of the bent thin film. We also show that the curved perovskite lasers are extremely robust with respect to repeated deformations. Because of the low spatial coherence, these curved random laser devices are efficient and durable speckle-free light sources for applications in spectroscopy, bioimaging, and illumination.