Suppose $X$ a normal project variety over complex numbers, $D$ is an integral Cartier divisor on $X$, and $L$ is a line bundle on $X$. Suppose one know the natural map $H^0(X, L) \to H^0(D, L|_D)$ is **surjective**, I want to know if the following statement holds:

For any effective divisor $G_D \in |L|_D|$, is there an effective divisor $G \in |L|$, such that $G|_D = G_D$? (Of course $G, D$ should not have common components in order to make sense of $G|_D$.)

Actually, my feeling is that this holds when there exists a divisor $R \in |L|$ such that $D$ is not a component of $R$, but may not hold in general. Hence I try to find a counterexample by looking at the case when $|L|$ has fixed component, and $D$ is one such fixed component. However, I don't have too much such example on hand.