Understanding magnetic-field generation and amplification in turbulent plasma is essential to account for observations of magnetic fields in the universe. A theoretical framework attributing the origin and sustainment of these fields to the so-called fluctuation dynamo was recently validated by experiments on laser facilities in low-magnetic-Prandtl-number plasmas ($mathrm{Pm} < 1$). However, the same framework proposes that the fluctuation dynamo should operate differently when $mathrm{Pm} gtrsim 1$, the regime relevant to many astrophysical environments such as the intracluster medium of galaxy clusters. This paper reports a new experiment that creates a laboratory $mathrm{Pm} gtrsim 1$ plasma dynamo for the first time. We provide a time-resolved characterization of the plasmas evolution, measuring temperatures, densities, flow velocities and magnetic fields, which allows us to explore various stages of the fluctuation dynamos operation. The magnetic energy in structures with characteristic scales close to the driving scale of the stochastic motions is found to increase by almost three orders of magnitude from its initial value and saturate dynamically. It is shown that the growth of these fields occurs exponentially at a rate that is much greater than the turnover rate of the driving-scale stochastic motions. Our results point to the possibility that plasma turbulence produced by strong shear can generate fields more efficiently at the driving scale than anticipated by idealized MHD simulations of the nonhelical fluctuation dynamo; this finding could help explain the large-scale fields inferred from observations of astrophysical systems.