The phrase "the narrative forward" is correct and usable in written English.
You can use it when referring to the development of a story or plot. For example, "The author used clever foreshadowing to drive the narrative forward."
"It's what's pushing the narrative forward," she ventured.
People suddenly appear, assume identities and move the narrative forward.
His clockwork, cause-and-effect plotting moves the narrative forward with inexorable logic and efficiency.
Instead of conventional storytelling, juxtaposition and association move the narrative forward.
In terms of plot, the character lives a fairly ordinary life, but the moments that create the dramatic tension and drive the narrative forward are poignant and resonant.
Angelic figures, motivated by what they call "a cold fascination with human disaster", propel the narrative forward and are incarnated within it.
In fact, far from propelling the narrative forwards, his compulsive chatter only slows things down, including his investigation into Minna's death.
With great skill, Yeston and Stone weave a series of love stories into the drama and, while you don't exactly come out humming the score, it drives the narrative forwards and relies heavily on the English choral tradition of Elgar and Vaughan Williams.
Both the President and the Administration don't spend a lot of time pushing that narrative forward.
Danny Duncan of the National Catholic Reporter particularly praised the performances of Belzer, Beatty and Kotto, as well as the police relationship between Pembleton and Bayliss, which he said serves as the source of dramatic tension driving the series narrative forward in the early episodes of the first season.
Though the source of this stereotype is elusive and certainly not singular, the Faces of Meth campaign, which began in a county in Oregon in 2004, is one piece that majorly pushed this narrative forward.
