In this paper, we study some properties of retraction sequences, blowups of polytopes and their interrelations. We introduce the blowups of quasitoric orbifolds using the combinatorial data associated with its orbit spaces. We prove that neither new singularity nor new torsion arises in the integral cohomologies of certain blowups of a quasitoric orbifold. Moreover, we show that if a quasitoric orbifold has no p-torsion in its cohomology then the cohomology of its blowup along a fixed point has no p-torsion. As a consequence, we construct infinitely many integrally equivariantly formal quasitoric orbifolds from a given one. We also investigate the torsions in the cohomologies of blowups of a class of simplicial toric varieties. © 2021 Elsevier B.V.