The phrase "based on popular demand" is correct and can be used in written English. You can use it to describe a decision or action that was taken due to a large number of people asking for it. For example: "The restaurant has added breakfast to the menu, based on popular demand.".
Based on popular demand, we're bringing the game room back to the after-party.
Ron, Tim, and Robert were experts last year and have returned based on popular demand.
As with Uber, the plans are to add these new routes based on popular demand from its members.
Yossi, Sean and Kevin were experts last year and return based on popular demand with Roelof Botha, Marc Andreessen and Marissa Mayer.
Last Wednesday, I met with an executive from Demand Media, a company that generates content based on popular Web searches and other data.
Films based on popular toys are a dime a dozen.
Herein, some new products based on popular warm-weather cocktails.
I love the desktop app, it’s always running on my Mac. Ludwig is the best English buddy, it answers my 100 queries per day and stays cool.
Cristina Valenza
Retail Lead Linguist @ Apple Inc.