[susiedqq]

He drew pictures of his bag on a piece of paper with lines.

To me, this simply means that he used lined paper (a paper pre-printed with lines - which is not good for drawing, by the way!)

So you could say, "on a piece of lined paper."

However, if you want to say his drawing was composed of lines, then place "with lines" near the word "drew" to indicate "how" he drew.
He drew pictures of his bag with lines.
"He drew with lines pictures of his bag" is awkward.

But this could also mean the bag was decorated with lines.
He drew pictures of his bag with lines.