In this paper, we focus on the three-dimensional growth of the vortical structures behind two square cylinders when placed in a side-by-side arrangement and examine different possible modes of flow bifurcation along their spanwise extended corelines. For this, unsteady three-dimensional flow simulations are conducted with five different values of gap/diameter ratios (g*), namely, 2.1, 1.7, 0.7, 0.5, and 0.2, that cover important known phases, i.e., coshedding (g* = 2.1 and 1.7), asymmetric (g* = 0.7 and 0.5), and single-body type (g* = 0.2) of wake evolution. In order to exploit flow physics within the transition range, the Reynolds number of the flow is taken as 100. Notably, for all the investigated cases, parallel vortex shedding has been observed behind the two cylinders. However, with the decrease of gap ratio g*, the downstream flow gradually lost stability and the corelines of the shedded vortices appeared in a wavy fashion. While for g* = 2.1 and 1.7, the simulated streamwise flows exhibit antiphase and in-phase synchronization of the wake, respectively, there is also observed a notable difference in the structural growth of corresponding spanwise flows. For the antiphase flow (at g* = 2.1) the downstream wake evolved through a symmetry-breaking mode-I Hopf bifurcation along various spanwise extended vortex corelines. However, the presence of an additional mode-II Hopf bifurcation was detected in the wake of an in-phase flow (at g* = 1.7). The development of a number of local pressure maxima over different spanwise extended vortex corelines, and the gradual decrease of pressure along their left and right, respectively, are noted to be responsible for initiating such flow bifurcations. In the biased (asymmetric) regime, with g* = 0.7 and 0.5, the flow behind the cylinders showed visible signs of growing instability and there occurred significant spanwise pressure fluctuations, which in turn inflicted large three dimensionality into the downstream wake and local velocity irregularities. The corresponding wake evolution patterns are thereby noted to exhibit predominantly unsteady and transitional features. In this biased flow regime, the gap flow flip-flopped randomly and became deflected alternately towards the top and bottom cylinders. On the one hand, clear dominance of both mode-I and mode-II flow bifurcations was observed with both of these initially locked biased flows. However, due to increased flow asymmetry the spanwise length scales of such bifurcations changed significantly. On the other hand, for the single-body regime flow with g* = 0.2, the occurrence of only mode-I bifurcation was detected in the wake of the cylinders.