You can consider drown out (or drown) and it fits to your example sentence also. Though, it can be used in situations where any loud sound overwhelms another sound.
For example, a very loud sound (a passing train) can make another sound (your voice) inaudible. In this case, the sound of the train drowns out your voice (or the train drowns you out). But, you can shout to drown out other sounds also just to be heard, it is not always making the other sound completely inaudible.
I had to shout so as to drown out the loudness of the music.
Examples from Google Books:
He was also speaking louder to drown out the sirens that suddenly seemed to be coming from everywhere, and because he knew Caleb would say they needed to run.
[The Harris Family: A Novel By RM Johnson (2012)]
"What are you two doing home?" Noelle yelled to drown out the music.
[Fierce Overture By Gun Brooke (2010)]
If you want to specifically indicate shouting, you can consider shout down. It is used in situations where you overwhelm someone or something by shouting (or talking loudly) to be heard.
The lecturer had to shout down the entire audience to be heard.
shout down verb (transitive, adverb) to drown, overwhelm, or silence by shouting or talking loudly