Let me assume everything in sight is as nice as possible, probably if the result I want is true then these conditions are too restrictive. All spaces will be smooth algebraic varieties over the complex numbers. We are given maps $j \colon U \to X$, $g \colon X \to S$, $f = g\circ j$. The map $j$ is an open immersion whose complement is a simple normal crossing divisor, $g$ is a smooth projective morphism, and $f$ is topologically a locally trivial fibration.

On one hand, we can restrict cohomology classes on $X$ to $U$ fiberwise, giving us $$ \newcommand{\Q}{\mathbf{Q}}R^qg_\ast \Q \twoheadrightarrow W_qR^qf_\ast\Q \hookrightarrow R^qf_\ast\Q.$$ Here $W_\bullet$ denotes the weight filtration on $R^qf_\ast\Q$, considered as a variation of mixed Hodge structure.

On the other hand, one can also consider $$ H^\bullet(X,\Q) \twoheadrightarrow \mathrm{Im}(j^\ast) \hookrightarrow H^\bullet(U,\Q)$$ given by restricting cohomology classes globally.

Question 1: Is there a "Leray" spectral sequence $H^p(S,W_qR^qf_\ast\Q) \implies \mathrm{Im}(j^\ast)$, compatible with the maps above and the Leray spectral sequences for $f$ and $g$?

Question 2: If so, does it always degenerate at $E_2$, like the Leray spectral sequence for $g$?