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We show that the topological Kitaev spin liquid on the honeycomb lattice is extremely fragile against the second-neighbor Kitaev coupling K_{2}, which has recently been shown to be the dominant perturbation away from the nearest-neighbor model in iridate Na_{2}IrO_{3}, and may also play a role in α-RuCl_{3} and Li_{2}IrO_{3}. 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 Na_{2}IrO_{3}. Moreover, the minimal K_{1}−K_{2} 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.