We measured by laser spectroscopy the isotope shifts between naturally occurring even isotopes of strontium ions for both the $5s{}^2{S}_{1/2}\to 5p{}^2{P}_{1/2}$ (violet) and the $4d{}^2{D}_{3/2}\to 5p{}^2{P}_{1/2}$ (infrared) dipole-allowed optical transitions. Fluorescence spectra were taken by simultaneous measurements on a two-component Coulomb crystal in a linear Paul trap containing ${10}^3$–${10}^4$ laser-cooled ${\mathrm{Sr}}^{+}$ ions. The isotope shifts are extracted from the experimental spectra by fitting the data with the analytical solution of the optical Bloch equations describing a three-level atom interacting with two laser beams. This technique allowed us to increase the precision with respect to previously reported data obtained by optogalvanic spectroscopy or fast atomic-beam techniques. The results for the $5s{}^2{S}_{1/2}\to 5p{}^2{P}_{1/2}$ transition are ${\nu }_{88}-{\nu }_{84}=+378\left(4\right)$ MHz and ${\nu }_{88}-{\nu }_{86}=+170\left(3\right)$ MHz, in agreement with previously reported measurements. In the case of the previously unexplored $4d{}^2{D}_{3/2}\to 5p{}^2{P}_{1/2}$ transition we find ${\nu }_{88}-{\nu }_{84}=-828\left(4\right)$ MHz and ${\nu }_{88}-{\nu }_{86}=-402\left(2\right)$ MHz. These results provide more data for stringent tests of theoretical calculations of the isotope shifts of alkali-metal-like atoms. Moreover, they simplify the identification and the addressing of ${\mathrm{Sr}}^{+}$ isotopes for ion frequency standards or quantum-information-processing applications in the case of multi-isotope ion strings.

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