We report the influence of epitaxial strain on the multiferroic phase transitions of BiFeO3 films. Using advanced characterization techniques and calculations we show that while the magnetic Néel temperature hardly varies, the ferroelectric Curie temperature TC decreases dramatically with strain. This is in contrast with the behavior of standard ferroelectrics where strain enhances the polar cation shifts and thus TC. We argue that this is caused by an interplay of polar and oxygen tilting instabilities and that strain can drive both transitions close together to yield increased magnetoelectric responses.