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 S-1(0) -> P-3(1) and S-1(0) -> P-3(0) transitions between Sr-84, Sr-86, and Sr-87, relative to Sr-88. Using the isotope-shift measurements of the two transitions, a King plot analysis is performed, revealing a nonlinearity in the measured values.