In this talk, we present a strong form of the quantitative fractional isoperimetric inequality. In particular, we show that the square root of the fractional isoperimetric deficit controls not only the Fraenkel asymmetry, but also a fractional-order oscillation of the boundary. This result can be viewed as the nonlocal counterpart of the local inequality established by Fusco and Julin. Our proof strategy relies on a regularization procedure for a volume-constrained shape-optimization problem driven by a nonlocal energy functional presenting both attractive and repulsive interactions. As an application, we will also discuss new stability estimates for a fractional Cheeger inequality. This is joint work with E. Cinti and B. Ruffini.