The index of refraction plays a decisive role in the design and classification of optical materials and devices; therefore, its proper and accurate determination is essential. In most refractive index (RI) sensing schemes, however, there is a trade-off between providing high-resolution measurements and covering a wide range of RIs. We propose and experimentally demonstrate a novel mechanism for sensing the index of refraction of a medium by utilizing the orbital angular momentum (OAM) of structured light. Using a superposition of co-propagating monochromatic higher-order Bessel beams with equally spaced longitudinal wavenumbers, in a comb-like setting, we generate non-diffracting rotating light structures in which the orientation of the beam's intensity profile is sensitive to the RI of the medium (here, a fluid). In principle, the sensitivity of this scheme can exceed ~2700°/RI unit (RIU) with a resolution of ~ RIU. Furthermore, we show how the unbounded degrees of freedom associated with OAM can be deployed to offer a wide dynamic range by generating structured light that evolves into different patterns based on the change in RI. The rotating light structures are generated by a programmable spatial light modulator. This provides dynamic control over the sensitivity, which can be tuned to perform coarse or fine measurements of the RI in real time. This, in turn, allows high sensitivity and resolution to be achieved simultaneously over a very wide dynamic range, which is a typical trade-off in all RI sensing schemes. We thus envision that this method will open new directions in refractometry and remote sensing.