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The conclusion is not true. Let $p > 1 ,$ $A = [ 0 , 1 ]$ and $f_j( x ) = j ( \log j {)}^{- 1 }$for $0 \le x \le j^{- 1 }$and $f ( x ) = 0$ otherwise. Then $\Vert f_j{\Vert }_1\to 0.$ On the other hand
$\Vert f_j{\Vert }_p^p= j^{p - 1 }( \log j {)}^{- p }\to \mathrm{\infty } .$
As $f_j$are not bounded in $L^p,$ in view of the Banach-Steinhaus theorem, there is $g \in L^q$such that the sequence
$\underset{0}{\overset{1}{\int }}{f}_j( x ) g ( x ) d x$
is unbounded.

Remark The conclusion is supported by the additional supposition that $\Vert f_j{\Vert }_p$are uniformly bounded, say by $M .$ Then we can approximate $g \in L^q$by combining indicators in a linear fashion. Indeed, for $g \in L^q$and $N > 0$ let
$A_N= \{ x \in A : | g ( x ) | \le N \}$
Then
$\underset{N}{lim}\underset{A \setminus A_N}{\int }| g ( x ) {| }^{q}d x = 0.$
For a given $\epsilon > 0$ there is $N$ such that
$\underset{A \setminus A_N}{\int }| g ( x ) {| }^{q}d x < \frac{1}{2}{\epsilon }^q$
For $n \in \mathbb{N}$and $1 \le k \le n$ let
$B_{k , n }= \{ x \in A : \frac{( k - 1 ) N }{n}< | g ( x ) | \le \frac{k N }{n}\}$
Define
$g_n( x ) = \sum _{k = 1 }^n\frac{k N }{n}1 {\text{I} }_{B_{k , n }}$
Next
$$\begin{gathered} \Vert g_n- g {\Vert }_q^q= \underset{A}{\int }| g_n( x ) - g ( x ) {| }^{q}d x = \underset{A_N}{\int }| g_n( x ) - g ( x ) {| }^{q}d x + \underset{A \setminus A_N}{\int }| g ( x ) {| }^{q}d x \\ = \sum _{k = 1 }^n\underset{B_{k , n }}{\int }| g_n( x ) - g ( x ) {| }^{q}d x + \frac{1}{2}{\epsilon }^q\le \frac{N^q}{n^q}\sum _{k = 1 }^{n}m ( B_{k , n }) + \frac{1}{2}{\epsilon }^q\le \frac{N^q}{n^q}m ( A ) + \frac{1}{2}{\epsilon }^q \end{gathered}$$
Fix $n$ such that ${\displaystyle \frac{N^q}{n^q}m ( A ) \le \frac{1}{2}{\epsilon }^q. }$Then $\Vert g_n- g {\Vert }_q\le \epsilon .$
Next
$\underset{A}{\int }{f}_{j}g d x = \underset{A}{\int }{f}_j[ g - g_n] d x + \underset{A}{\int }{f}_j{g}_{N}d x$
Consequently
$$\begin{gathered} \left| \underset{A}{\int }{f}_{j}g d x \right| \le \underset{A}{\int }| f_j| | g - g_n| d x + \left| \underset{A}{\int }{f}_j{g}_{n}d x \right| \\ \le \Vert f_j{\Vert }_p\Vert g - g_n{\Vert }_q+ \left| \underset{A}{\int }{f}_j{g}_{n}d x \right| \le M \epsilon + \left| \underset{A}{\int }{f}_j{g}_{n}d x \right| \end{gathered}$$
Using assumptions, we
$\underset{j}{lim}\underset{A}{\int }{f}_j{g}_{n}d x = 0$
The conclusion is thus drawn.