Bulk and surface state contributions to the electrical resistance of single-crystal samples of the topological Kondo-insulator compound SmB_{6} are investigated as a function of crystal thickness and surface charge density, the latter tuned by ionic liquid gating with electrodes patterned in a Corbino disk geometry on a single (100) surface. By separately tuning bulk and surface conduction channels, we show conclusive evidence for a model with an insulating bulk and metallic surface states, with a crossover temperature that depends solely on the relative contributions of each conduction channel. The surface conductance, on the order of 100 e^{2}/h, exhibits a field-effect mobility of 133 cm^{2}/Vs and a large carrier density of ∼2×10^{14} cm^{-2}, in good agreement with recent photoemission results. With the ability to gate modulate surface conduction by more than 25%, this approach provides promise for both fundamental and applied studies of gate-tuned devices structured on bulk crystal samples.