We present near-IR narrowband continuum-subtracted images in the H2 2.122 μm, and Brγ 2.166 μm emission lines for a sample of 15 bipolar planetary nebulae (PNe). H2 emission was definitely detected for most of the objects in this sample (13 of 15). The very high H2 detection rate supports the idea that bipolar PNe have important reservoirs of molecular material and offer suitable physical conditions for the excitation of H2. The strength of the H2 emission and the H2/Brγ flux ratio are found to correlate with the morphology of the bipolar PNe observed. Bipolar PNe with broad and bright rings exhibit stronger H2 emission than bipolar PNe with narrow twists. High-quality (subarcsec) [N II] and Hα optical images have been used to compare the distribution of the ionized and molecular material. The H2 emission lies just outside the optical [N II] emission zone.