We prove the operator-space Grothendieck inequality for bilinear forms on subspaces of noncommutative $L_p$-spaces with $2<p<\infty$. One of our results states that given a map $u:E\to F^{*}$, where $E,F\subset L_p(M)$ ($2<p<\infty$, $M$ being a von Neumann algebra), $u$ is completely bounded if and only if $u$ factors through a direct sum of a $p$-column space and a $p$-row space. We also obtain several operator-space versions of the classical little Grothendieck inequality for maps defined on a subspace of a noncommutative $L_p$-space ($2<p<\infty$) with values in a $q$-column space for every $q\in [p\text{'},p]$ ($p\text{'}$ being the index conjugate to $p$). These results are the $L_p$-space analogues of the recent works on the operator-space Grothendieck theorems by Pisier and Shlyakhtenko. The key ingredient of our arguments is some Khintchine-type inequalities for Shlyakhtenko's generalized circular systems. One of our main tools is a Haagerup-type tensor norm that turns out to be particularly fruitful when applied to subspaces of noncommutative $L_p$-spaces ($2<p<\infty$). In particular, we show that the norm dual to this tensor norm, when restricted to subspaces of noncommutative $L_p$-spaces, is equal to the factorization norm through a $p$-row space