The phrase "avoid bringing up" is correct and usable in written English.
It can be used when advising someone to refrain from mentioning a particular topic or subject.
Example: "In our meeting, let's avoid bringing up past conflicts to keep the discussion productive."
Alternatives: "steer clear of mentioning" or "refrain from discussing".
That makes it all the more likely that Democratic candidates will avoid bringing up the issue with voters.
Separate bank accounts prevent her from having to see the money spent on annual dues, and the two manage to avoid bringing up the subject at home.
One Jewish leader in the state said, to get along with him, friends just had to remember who was feuding with whom and avoid bringing up that name.
Importantly, this research is not intended to suggest that people — men or women — should speak up only with ideas and avoid bringing up problems.
While activists for and against the pipeline gird for a highly public fight, there are many others here who find some merit in each side's arguments, and who mostly avoid bringing up Keystone XL in polite conversation.
If I can avoid bringing up that fact at all, I avoid it.
I understood my colleagues' hesitation because I always avoided bringing up my own decision to take a professional hiatus.
Ms. Mobaslat, of Save the Children, said aid workers with her group avoided bringing up sectarian feelings directly because they tried not to start conversations they could not resolve.
In the past, Dev has avoided bringing up his kink with lovers who had body image issues with their armpits.
Kim returned the favor when North Korea backed Trump for president, although the Republican avoided bringing up that particular endorsement during the debates.
Her voice is friendly, somewhat placid, but it's clear Romero is not thrilled with having to answer questions about her political alliance with the Koch brothers and other wealthy supporters of Proposition 32, and she conspicuously avoids bringing up their names.
