We study the (intrinsic) biharmonic map heat flow on a four-dimensional domain into a compact Riemannian manifold. We examine its behavior as the first finite-time singularity is approached. At each singular point, we find either a harmonic sphere or a biharmonic map bubbling-off. A further description of the singular set is also given. The proofs rely to a large extent on a blowup analysis of sequences of maps with uniformly bounded energy.