We prove the operator-space Grothendieck inequality for bilinear forms on subspaces of noncommutative $L_p$ -spaces with $2 \lt p \lt \infty$ . One of our results states that given a map $u: E\to F^*$ , where $E, F\subset L_p(M)$ ( $2 \lt p \lt \infty$ , $M$ 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 ( $2 \lt p \lt \infty$ ) with values in a $q$ -column space for every $q\in [p', p]$ ( $p'$ 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 ( $2 \lt p \lt \infty$ ). 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