We propose and test a method to interpolate sparsely sampled signals by a stochastic process with a broad range of spatial and/or temporal scales. To this end, we extend the notion of a fractional Brownian bridge, defined as fractional Brownian motion with a given scaling (Hurst) exponent H and with prescribed start and end points, to a bridge process with an arbitrary number of intermediate and nonequidistant points. Determining the optimal value of the Hurst exponent H_{opt}, appropriate to interpolate the sparse signal, is a very important step of our method. We demonstrate the validity of our method on a signal from fluid turbulence in a high Reynolds number flow and discuss the implications of the non-self-similar character of the signal. The method introduced here could be instrumental in several physical problems, including astrophysics, particle tracking, and specific tailoring of surrogate data, as well as in domains of natural and social sciences.