Shopping malls across the nation were taking additional security precautions Tuesday following a string of disturbances the day after Christmas that resulted in minor injuries, evacuations and scores of arrests.

Police were beefing up patrols and mall security guards were out in force after melees involving mostly young people erupted around malls in Elizabeth, N.J., Fayetteville, N.C., East Garden City, N.Y., Aurora, Colo., and Tempe, Ariz., among others.

In Manchester, Conn., around 6 p.m. Monday several hundred teens began fighting at the Shoppes at Buckland Hills. Police from a half dozen communities rushed to the mall and arrested eight people, police said.

About the same time at the Fox Valley Mall in Aurora, Ill., a large group of teenagers in the food court began fighting, according to a spokesman for the police department there. A police sergeant and private guard on duty at the mall tried to break up the fight. When they couldn’t disperse the crowd, the teens started throwing things at them. They called in backup and 75 additional police from eight nearby departments rushed to the mall where they arrested eight teenagers, including five girls. Then police evacuated the mall and closed it for the night.

Around 5 p.m. at the Mills at Jersey Gardens in Elizabeth four young women eating in the food court got into a loud altercation when one of them picked up a chair and slammed it down. The noise prompted someone to scream “shots fired” or “gun,” Elizabeth Mayor Chris Bollwage said on Tuesday.

There were about 5,000 people in the mall at the time and the ensuing panic and rush to escape the area resulted in injuries to about 10 people, including a man who broke a leg, another man who cut his finger and two pregnant women who were pushed and jostled, according to the mayor.

Several people were treated by emergency medical technicians at the scene and taken to area hospitals, where they were treated and released.

Many of the merchants pulled their doors or grates closed during the panic, locking the employees and shoppers inside. Police were able to determine fairly quickly there was no gun or gun-related injuries but the mall was evacuated and closed early. It reopened Tuesday morning.

Mall operators are cooperating with law enforcement on investigations, according to the International Council of Shopping Centers.

“Malls are a place for communities to shop, eat, be entertained and come together, particularly during the holiday season,” the trade group said in a statement. “Consumers and shopping center employees can be assured that their security is the industry’s primary focus 365 days a year.”

Les Morris, a spokesman for Simon Property Group, said that the company has launched an investigation, including into the instance at the Mills at Jersey Garden, which it owns.

The incidences resemble the flash mobs of a few years ago where a few dozen young adults would use social media to coordinate robberies of a convenience stores, said Gus Downing, publisher of D&D Daily, an online newsletter that covers retail loss prevention and security. These “flash fights” can be traumatic because mall security guards and store workers use a similar protocol to that of an active shooter.

“You don’t know what’s going to evolve. One kid could have a gun and you’ve got to protect the public,” Mr. Downing said. “It’s psychologically traumatic for everyone.”

“If you look around the country it’s all young people gathering at the mall the day after Christmas,” Mayor Bollwage said. “I don’t believe it was a coordinated social-media flash mob kind of thing, I think it was just frustration at the end of the holidays and a climate of too many people in one area.”

Mr. Bollwage was eating in the food court on Tuesday along with other city officials to reinforce the message that the mall was safe and open for business. The mall, which had about 45,000 shoppers over the course of the day on Monday, provides a significant tax benefit to the struggling city of about 125,000. He said the police presence inside and outside the mall had roughly doubled from Monday and noted cameras captured the license plates of every car that entered the mall parking lot.

“If you look around the world nothing is getting solved through dialogue or conversation, issues around the world are being solved through violence,” Mr. Bollwage said. An incident like this “is a commentary on humanity at this stage.”

Write to Douglas Belkin at doug.belkin@wsj.com and Paul Ziobro at Paul.Ziobro@wsj.com