We establish convergence in norm and pointwise almost everywhere for the non-conventional (in sense of Furstenberg) bilinear polynomial ergodic averages $$ A_N(f,g)(x) := \frac{1}{N} \sum_{n=1}^{N} f(T^nx) g(T^{P(n)}x) as $N \to \infty$, where $T\colon X\to X$ is a measure-preserving transformation $\sigma$-finite measure space $(X,\mu), P(\mathrm{n})\in \mathbb{Z}[\mathrm{n}]$ degree $d \geq 2...