'gaining traction quickly' is correct and usable in written English.
You can use the phrase to describe something that is quickly gaining popularity or recognition, usually among a particular group or in a particular context. For example, "The new movie franchise has been gaining traction quickly among teenage moviegoers."
Samsung's new Android smartphone, the Galaxy S III, is gaining traction quickly.
As the episode comes to a close, Claire is in third place for the vice-presidential nomination but gaining traction quickly.
And the nifty service is gaining traction quickly amongst users.
Though Fandango is a well-entrenched player, Atom Tickets has been gaining traction quickly thanks to its partnerships with major movie studios, as well as companies like Amazon, Chase Pay and T-Mobile.
From their offices in New York City, they are gaining traction quickly, and are working to 'open the floodgates of credit' using a decentralized system similar to my description above, to build an ethereum-member based lending platform.
It gained traction quickly and drew a lot of attention to some of the family's recent activism.
TechArb, which is located just down the hall from the Google AdWords office in Ann Arbor, allows student startups to gain traction quickly, try out ideas, fail, pivot and try again.
Longtime patrons of the store rallied and the Cooper Square Committee, a local community organization, started an online petition called "Save St . Marks Bookshop From Cooper Union" that gained traction quickly.
Frederic Neumann, senior Asia economist at HSBC in Hong Kong, said the growth figures were a "welcome surprise," and showed that the government's fiscal stimulus was gaining traction more quickly than forecast.
However, the ability to quickly load interactive micro apps directly into messengers and even search results is quickly gaining traction.
