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Isotope-shift spectroscopy with narrow optical transitions provides a benchmark for atomic structure calculations and has also been proposed as a way to constrain theories predicting physics beyond the standard model. Here we measure frequency shifts of the ^{1}S_{0}→^{3}P_{1} and ^{1}S_{0}→^{3}P_{0} transitions between ^{84}Sr,^{86}Sr, and ^{87}Sr, relative to ^{88}Sr. Using the isotope-shift measurements of the two transitions, a King plot analysis is performed, revealing a nonlinearity in the measured values.