  Let $\Gamma$ be a finitely generated group and $N$ be a normal subgroup of
$\Gamma$. The fiber product of $\Gamma$ with respect to $N$ is the subgroup
$\Gamma \times_N \Gamma=\{(\gamma, \gamma w): \gamma \in \Gamma, w \in N\}$ of
the direct product $\Gamma \times \Gamma$. For every representation
$\rho:\Gamma \times_N \Gamma \rightarrow \mathsf{GL}_d(k)$, where $k$ is a
local field, we establish upper bounds for the norm of the Cartan projection of
$\rho$ in terms of a fixed word length function on $\Gamma$. As an application,
we exhibit examples of finitely generated and finitely presented fiber products
$P=\Gamma \times_N \Gamma$, where $\Gamma$ is linear and Gromov hyperbolic,
such that $P$ does not admit linear representations which are quasi-isometric
embeddings.
