Steep gradients of temperature and density, called cold fronts, are observed by Chandra in a leading edge of subclusters moving through the intracluster medium (ICM). The presence of cold fronts indicates that thermal conduction across the front is suppressed by magnetic fields. We carried out three-dimensional magnetohydrodynamic (MHD) simulations including anisotropic thermal conduction of a subcluster moving through a magnetically turbulent ICM. We found that turbulent magnetic fields are stretched and amplified by shear flows along the interface between the subcluster and the ambient ICM. Since magnetic fields reduce the efficiency of thermal conduction across the front, the cold front survives for at least 1 Gyr. We also found that a moving subcluster works as an amplifier of magnetic fields. Numerical results indicate that stretched turbulent magnetic fields accumulate behind the subcluster and are further amplified by vortex motions. The moving subcluster creates a long tail of ordered magnetic fields, in which the magnetic field strength attains a value of β = P_gas/P_mag 10.