The phrase "escape quickly" is correct and can be used in written English.
You can use it when you are describing the action of fleeing rapidly or when someone is trying to avoid a situation. For example: "The crowd began to push, so I had to escape quickly before I got trampled."
A fire crew spotted them and, with its ladders, helped the trapped people escape quickly.
He said some people near the bar and back exit managed to escape quickly.
-The need to escape quickly from a severed gas main is so clear that Entex employees are taught to formulate escape routes in advance.
For part-time and low-wage workers, the fees, which can be difficult to escape, quickly devour much of the money deposited on the cards.
She knew she had to escape quickly, even if she was hobbled by Cuthbert, her seven-pound wooden leg, which she had relied on after a hunting accident.
The Corvette he was driving exploded in flames when it hit a concrete wall, but Earnhardt was able to escape quickly enough to avoid serious injury.
Their atmospheric gases either escaped quickly to space from their weak gravitational fields or were blown off by major impacts.
Mr. Dudley and the front-seat passenger, Paul Nicholson, escaped quickly, but the three women who were in the rear seats did not survive.
A hydrogen flow of 10 dm3 min-1, measured at ameasuredemperature ambientssure, temperature reandor ensured that the overall conversion was not limited by mass transfer in the sampressure2] and throughlathees escaped quickly.
Some were freed or escaped quickly, but 125,000 were imprisoned in camps run by the NKVD.
The elevator, once a marvel of efficiency, has become a social purgatory from which most of us cannot escape too quickly.
