We investigate the stretching mechanism of Finitely Extensible Nonlinear Elastic (FENE) model of polymers in a random turbulent flow. The turbulent model includes a dominant space-scale $\ell\sim N^{-1}$, a dominant time-scale $τ$, and is white in time. Under suitable scaling assumption, the polymer density equation, initially a stochastic Fokker-Planck equation in the presence of transport-stretching noise, converges weakly as $N\uparrow \infty$ to a limit deterministic equation with a new extra term, a second order operator. This operator, whose shape has been predicted in the physical literature by other arguments, express a sort of average `turbulent stretching' effect. With respect to other derivation of this effective model, the main novelty of our approach is that the deterministic limit is obtained pathwise, without having to take averages with respect to different realizations of the random flow. Next, we consider the limit as $τ\downarrow 0$ and we identify the stationary distribution of the polymer length. The analysis is carried out in appropriate weighted spaces, which take into account the singularity of the FENE force near the boundary and the no-flux boundary condition, and combines stochastic scaling limit and singular limit techniques.