Tangential discontinuities, seen as x-ray edges known as cold fronts (CFs), are ubiquitous in cool-core galaxy clusters. We analyze all 17 deprojected CF thermal profiles found in the literature, including three new CFs we tentatively identify (in clusters A2204 and 2A0335). We discover small but significant thermal pressure drops below all nonmerger CFs, and argue that they arise from strong magnetic fields below and parallel to the discontinuity, carrying 10%-20% of the pressure. Such magnetization can stabilize the CFs, and explain the CF-radio minihalo connection.