  The recently discovered nickelate superconductor La$_3$Ni$_2$O$_7$ has a high
transition temperature near 80 K under pressure, which offers additional
avenues of unconventional superconductivity. Here with state-of-the-art
tensor-network methods, we study a bilayer $t$-$J$-$J_\perp$ model for
La$_3$Ni$_2$O$_7$ and find a robust $s$-wave superconductive (SC) order
mediated by interlayer magnetic couplings. Large-scale density matrix
renormalization group calculations find algebraic pairing correlations with
Luttinger parameter of $K_{\rm SC} \simeq 1$. Infinite projected entangled-pair
state method obtains a nonzero SC order directly in the thermodynamic limit,
and estimates a strong pairing strength $\bar{\Delta}_z \sim \mathcal{O}(0.1)$.
Tangent-space tensor renormalization group simulations further determine a high
SC temperature $T_c^*/J \sim \mathcal{O}(0.1)$ and clarify the temperature
evolution of SC order. Because of the intriguing orbital selective behaviors
and strong Hund's rule coupling in the compound, $t$-$J$-$J_\perp$ model has
strong interlayer spin exchange (while negligible interlayer hopping), which
greatly enhances the SC pairing in the bilayer system. Such a magnetically
mediated strong pairing has also been observed recently in the optical lattice
of ultracold atoms. Our accurate and comprehensive tensor-network calculations
reveal robust SC order in the bilayer $t$-$J$-$J_\perp$ model and shed light on
the high-$T_c$ superconductivity in the pressurized nickelate
La$_3$Ni$_2$O$_7$.
