Einstein's weak equivalence principle (WEP) states that any freely falling, uncharged test particle follows the same identical trajectory independent of its internal structure and composition. Since the polarization of a photon is considered to be part of its internal structure, we propose that polarized photons from astrophysical transients, such as gamma-ray bursts (GRBs) and fast radio bursts (FRBs), can be used to constrain the accuracy of the WEP through the Shapiro time delay effect. Assuming that the arrival time delays of photons with different polarizations are mainly attributed to the gravitational potential of the Laniakea supercluster of galaxies, we show that a strict upper limit on the differences of the parametrized post-Newtonian parameter. value for the polarized optical emission of GRB 120308A is Delta(gamma) < 1.2 x 10(-10), for the polarized gamma-ray emission of GRB 100826A is Delta(gamma) < 1.2 x 10(-10), and for the polarized radio emission of FRB 150807 is Delta(gamma) < 2.2 x 10(-6). These are the first direct verifications of the WEP for multiband photons with different polarizations. In particular, the result from FRB 150807 provides the most stringent limit to date on a deviation from the WEP, improving by one order of magnitude the previous best result based on Crab pulsar photons with different energies.