Mad was so successful that its publisher, EC, created its own knock-off called Panic. It wasn't edited by Harvey Kurtzman, though; it was edited by Al Feldstein. Here's the Wikipedia article abut Panic.
From Steve Stiles' article, "It's a Panic!":
What Panic also earned was a storm of indignation that burst over Gaines' head with the very first issue, and all over the holiday of "Peace on Earth, Good Will Towards Men." It's strange that Gaines didn't see it coming, but some people got very annoyed with a satire of "The Night Before Christmas." To put it mildly.
The lead story was provocative enough — an eight-page satire, drawn by Jack Davis, of crime novelist (and comics writer) Mickey Spillane's work, titled "My Gun Is the Jury." Feldstein set about to parody Mike Hammer's rather gory crime solving techniques, labeling the satire "Sex and Sadism Department." In page after page "Mike Hammershlammer" blows away a variety of beautiful women ("I let her have it, right in the gut, a little below the belly-button…"). In the last few panels Mike has shot Stella –only to discover that "she" is a "he." Not only that, but Mike "himself" is a female transvestite! That was pretty heavy stuff for the 1950s and came to cause Gaines considerable grief, as did the last story in the first issue!
Incidentally, Feldstein took over Mad after Hugh Hefner lured Kurtzman away from Mad by offering him his own humor magazine called Trump. It folded after three issues.
You can download scans of all 12 issues of Panic. (They're in the cbr format — download a reader here.)