We show that the topological Kitaev spin liquid on the honeycomb lattice is extremely fragile against the second-neighbor Kitaev coupling K₂, which has recently been shown to be the dominant perturbation away from the nearest-neighbor model in iridate Na₂IrO₃, and may also play a role in α-RuCl₃ and Li₂IrO₃. This coupling naturally explains the zigzag ordering (without introducing unrealistically large longer-range Heisenberg exchange terms) and the special entanglement between real and spin space observed recently in Na₂IrO₃. Moreover, the minimal K₁ - K₂ model that we present here holds the unique property that the classical and quantum phase diagrams and their respective order-by-disorder mechanisms are qualitatively different due to the fundamentally different symmetries of the classical and quantum counterparts.