"exert pressure upon" is correct and usable in written English.
You can use it to mean to use pressure or influence to bring about a desired result. For example: "The board used their power and influence to exert pressure upon the company to change their policies."
Although many human activities exert pressures upon wildlife, the magnitude of the potential impacts estimated for European breeding birds emphasises the importance of climatic change.
Furthermore, there's a suspicion, albeit perhaps groundless, that Perez has been exerting pressure upon his manager, whether explicit or implicit, to play Bale from the start, even though the Welshman is clearly less than fit following a disrupted summer involving no pre-season and a doubtless related series of irritating injuries.
Through her works, Cisneros conveys the experiences of Chicanas confronting the "deeply rooted patriarchal values" of Mexican culture through interactions not only with Mexican fathers, but the broader community which exerts pressure upon them to conform to a narrow definition of womanhood and a subservient position to men.
In the meantime, and as a lever to exert pressure, shop stewards embarked upon a broader political campaign for the right of all people to socially useful production.
Therefore, the onus will predictably fall upon China to exert pressure over its estranged ally.
We have to exert pressure gradually".
Between elections, citizens exert pressure to receive private benefits.
In "Talking Our Way to Peace" (Op-Ed, Dec. 2), former Secretary of State James A. Baker III discusses what Israel should do "upon the election of a Palestinian negotiating partner" and how America can exert pressure on Israel to facilitate peace.
