The phrase "almost overturned" is correct and usable in written English.
It can be used to describe a situation where something was nearly reversed or changed, often in a legal or decision-making context.
Example: "The court's decision was almost overturned by the new evidence presented during the appeal."
Alternatives: "nearly reversed" or "close to being annulled".
"Washington must be grinding its teeth because Snowden's revelations have almost overturned the image of the U.S. as the defender of a free Internet," Global Times, which often reflects the official point of view, wrote in an editorial.
Ottawa hasn't been this chaotic since Stephane Dion, Jack Layton, and Gilles Duceppe almost overturned the government.
There have been numerous reports recently about overturned convictions.
Even on the rare occasions when the parole board granted a release, California's two previous governors — Gray Davis, a Democrat, and Arnold Schwarzenegger, a Republican — almost invariably overturned it.
The Lancashire Evening Post picks up on Westminster rumours that Jack Straw's Blackburn seat has been put on the party's "at risk" list, saying his staunch support for the war in Iraq could even see his majority of almost 10,000 overturned.
After a year in which federal courts have almost unanimously overturned marriage bans, the Sixth Circuit has now become the first federal appellate court to uphold such a ban.
Three young men whose last meal was almost certainly noodle-based sit on overturned milk crates near a turntable.
A tabby crawls out of an overturned refrigerator near the edge of the forest.
While Jason comforts the dying Créuse as the two sing a poignant duet, the children are led to two of the gold boxes, now overturned to form coffins, and almost as if they were still playing, as we had witnessed them do earlier, they gently climb in.
On average almost two in five (38%) challenged decisions are overturned at tribunal, nearly one in 10 of all those made.
Near the roots of an overturned tree was a small plastic bucket.
