The phrase "outsize influence" means having a very strong and significant impact that is larger or more powerful than expected or necessary. Full definition
Price, who at first agreed to an interview but later stopped responding to messages, is only one of a number of private lawyers who have exerted outsize influence on American policy on ISDS. (buzzfeed.com)
-LSB-...] outsize influence of corporate money was on full display this week, as frozen pizza manufacturers were able to circumvent proposed changes to school food regulations -LSB-...] (thelunchtray.com)
Because committee members» ballots are weighted according to how many people in their area voted for the Democratic candidate in the most recent gubernatorial election, Upper West Siders wielded outsize influence in the decision process because of vastly higher turnout rates in their neighborhood — even though it makes up a relatively small portion of the district. (observer.com)