A temperature-insensitive high-sensitivity refractive index sensor is proposed and experimentally demonstrated, which is based on utilization of a thinned helical fiber grating but with an intermediate period (THFGIP). Attributed to the reduced diameter and an intermediate period of the grating, the proposed sensor has a high surrounding refractive-index (SRI) sensitivity and a low temperature sensitivity. The average SRI sensitivity of the proposed sensor is up to 829.9 nm/RIU in the range of 1.3410-1.4480 RIU. Moreover, unlike the traditional sensitivity-enhancement method by increasing the waveguide dispersion factor, here the waveguide dispersion factor at the resonant wavelength was decreased by reducing the diameter of the fiber grating and as a result, the crosstalk effect due to the temperature change can be further suppressed. The proposed temperature-insensitive SRI sensor has the superiorities of simple structure, ease fabrication, and low cost, which could be found more potential applications in the SRI sensing fields.