Shock waves appear in numerous high-speed propulsion applications, including intakes, nozzles, and transonic and supersonic turbomachinery. The aerodynamic performance in bladeless turbines, which is designed for work extraction under such conditions, is dominated by flow separation induced by shock-wave pressure gradients. The large velocity gradients pose limitations on flowfield characterization using particle-based optical diagnostics, such as particle image velocimetry and laser Doppler anemometry. These limitations, along with challenges in seeding the flow, can be overcome by tracking the molecules already present in the flow. This paper presents kHz-rate femtosecond laser electronic excitation tagging (FLEET) to excite long-lived fluorescence of nitrogen molecules, acting as in-situ flow tracers. A multi-point variation of this approach was demonstrated in an optically accessible linear turbine test section, developed to investigate bladeless turbines. The femtosecond laser is coupled with an intensified CMOS camera with a frame rate of 200 kHz. High-speed measurements were made of the steady and unsteady performance in the bladeless turbine, with particular attention to capturing flow structures, spatial velocity gradients, and transient events such as unstarting of the supersonic passages.