"lightning storm" is correct and usable in written English.
You can use it to describe a storm with a lot of lightning. For example, "A powerful lightning storm swept through the town last night."
China today is a train rushing through a lightning storm….
All it needs is a violent lightning storm.
"There was a magnificent lightning storm," Mr. Gordon said.
That game was postponed by a lightning storm.
If there's a lightning storm, I'm running away from this kid, 'cause he's cursed.
On Tuesday morning, a lightning storm set in and Nyad was pulled from the sea.
Look at the terrifying lightning storm, and the splendid corgi with his natural long tail.
The rest of the Islanders plopped into their couches and let the Lightning storm the crease.
"Yippee!" she shouted to a lightning storm of camera flashes and complaints from the other men.
As I leave the Nightcrawler press screening, a lightning storm hits Toronto.
In a lightning storm, this house will be the perfect place to be".
