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We prove rigorous bounds on the growth of $α$-Renyi entropies $S_α(t)$ (the Von Neumann entropy being the special case $α= 1$) associated with any subsystem $A$ of a general lattice quantum many-body system with finite onsite Hilbert space dimension. For completely non-local Hamiltonians, we show that the instantaneous growth rates $|S'_α(t)|$ (with $α\neq 1$) can be exponentially larger than $|S'_1(t)|$ as a function of the subsystem size $|A|$. For $D$-dimensional systems with geometric locality, we prove bounds on $|S'_α(t)|$ that depend on the decay rate of interactions with distance. When $α= 1$, the bound is $|A|$-independent for all power-law decaying interactions $V(r) \sim r^{-w}$ with $w > 2D+1$. But for $α> 1$, the bound is $|A|$-independent only when the interactions are finite-range or decay faster than $V(r) \sim e^{- c\, r^D}$ for some $c$ depending on the local Hilbert space dimension. Using similar arguments, we also prove bounds on $k$-local systems with or without geometric locality. A central theme of this work is that the value of $α$ strongly influences the interplay between locality and entanglement growth. In other words, the Von Neumann entropy and the $α$-Renyi entropies cannot be regarded as proxies for each other in studies of entanglement dynamics. We compare these bounds with analytic and numerical results on Hamiltonians with varying degrees of locality and find concrete examples that almost saturate the bound for non-local dynamics.