We investigate the chiral phase structure of three flavor QCD in a background $U\left(1\right)$ magnetic field using the standard staggered action and the Wilson plaquette gauge action. We perform simulations on lattices with a temporal extent of $N_{\tau }=4$ and four spatial extents of $N_{\sigma }=8$, 16, 20 and 24. We choose a quark mass in lattice spacing as $am=0.030$ with corresponding pion mass estimated as $m_{\pi }\sim 280\mathrm{MeV}$ such that there exists a crossover transition at vanishing magnetic fields, and adopt two values of magnetic field strength in lattice spacing $a\sqrt{eB}\simeq 1.5$ and 2 corresponding to $eB/m_{\pi }^2\sim 11$ and 20, respectively. We find that the transition becomes stronger in the presence of a background magnetic field, and turns into a first order as seen from the volume scaling of the order parameter susceptibility as well as the metastable states in the time history of the chiral condensate. On the other hand, the chiral condensate and transition temperature always increase with $B$ even within the regime of a first order phase transition. This suggests that the discrepancy in the behavior of chiral condensates and transition temperature as a function of $B$ between earlier lattice studies using larger-than-physical pion masses with standard staggered fermions and those using physical pions with improved staggered fermions is mainly due to lattice cutoff effects.