Optical molecular sensing is one of the most promising techniques for a wide range of applications, from fundamental studies to medical and environmental analysis. In the optical spectrum, the mid-infrared is particularly important for sensing because many molecules exhibit strong absorption features in this spectral region. To enhance the sensitivity of optical molecular sensing, typically passive and sometimes active cavities are used; however, these cavity-enhanced sensing techniques have so far faced fundamental trade-offs between sensitivity and dynamic range and practical challenges associated with the required cavity finesse and availability of laser gain materials. Here, we show that nonlinear dynamics in low-finesse resonators, namely the formation of quadratic cavity solitons, offer an enhancement mechanism for molecular sensing, which is not limited by these constraints. In proof-of-principle measurements of CO2, we show an equivalent path length enhancement of nearly 2500. Additionally, we demonstrate large dynamic range through measurement of high concentrations of CO2 with sensitivities that are orders of magnitude higher than those achievable through linear cavity-enhanced sensing, thereby breaking the fundamental limitations of standard optical sensing techniques.