Rivers rose rapidly across much of the Midwest and the South on Saturday, prompting water rescues, evacuation orders and road closures as a relentless storm dumped rain on the region.

The increased flooding, which was happening from Texas to Ohio, came after days of heavy rains and tornadoes that killed at least 16 people, including a 5-year-old in Arkansas and a Missouri firefighter. Forecasters warned that the floods might continue well into next week, with rivers not expected to crest in some places until Tuesday or Wednesday.

“We’ll be dealing with the river flooding the next couple days, even the next couple weeks in some places,” said Colby Pope, a meteorologist with the National Weather Service in Little Rock, Ark.

Emergency workers reported water rescues in Arkansas, Kentucky, Missouri and Texas. Officials in Kentucky issued evacuation orders for two small towns, Butler and Falmouth, along the Licking River, and for the city of Shelbyville, along the Big Blue and Little Blue Rivers. Residents along the Kentucky River have also been evacuated.

On Saturday, the storm battered northwestern Alabama, where a tornado touched down near the city of Florence, the National Weather Service said. And the agency issued flash flood emergencies for Memphis and the Little Rock area.

The storm has killed people across four states, including a 9-year-old boy who was swept away by floodwaters in Frankfort, Ky., and Chevy Gall, a firefighter with the Beaufort-Leslie Fire Protection District in Missouri, who died in a crash while driving to help rescue people from the floods. At least nine weather-related deaths have been reported in Tennessee.

Some of the heaviest rain so far has fallen in Arkansas and in southern Missouri. In Lonsdale, Ark., firefighters said on Saturday that they were trying to help 10 people and 65 horses evacuate a flooded horse training center. In West Plains, Mo., Mayor Mike Topliff said that some buildings had taken on water after several inches of rain fell in a few hours. He said there were nine water rescues in his city, and at least one person had died.

Across Arkansas, where trees and power lines had been toppled, and where a train derailed when a bridge collapsed, many residents were hunkering down. In Cabot, in rural Central Arkansas, some homes had become small islands, surrounded by floodwaters on all sides. More than 100,000 customers were without power in the state as of Saturday evening, according to poweroutage.us.

In Jacksonville, not far from Little Rock, Tonya Coosenberry stood on her front porch wrapped in a blanket, watching the road in front of her house disappear beneath the water. “This is probably the worst flooding we’ve had in nearly 15 years,” she said.

On Saturday, wet weather stretched from East Texas to New York, and federal data showed rivers rising rapidly in some parts of the region.

The Black River near Poplar Bluff, Mo., surged from four feet to a flood stage of nearly 17 feet overnight. The river is expected to continue rising there through Sunday morning, with an evening crest expected just below record levels. Emergency responders, including an urban search and rescue team, were waiting in Poplar Bluff on Saturday, preparing to help evacuate people if needed.

Elsewhere, water levels were increasing quickly on the Kaskaskia River in Illinois, the Mississippi River along the Missouri-Kentucky border and the Ohio River along the Illinois-Kentucky border. Many highways have closed in Indiana and Ohio, though some rivers there are not expected to reach peak levels for several more days. The Spring River in Hardy, Ark., was already at a major flood stage on Saturday, reaching the second-highest crest ever recorded there.

The ground is saturated and can no longer absorb the rain, which means it “has nowhere to go and it runs off and creates more flooding,” said Frank Pereira, a meteorologist at the Weather Prediction Center.

The stormy weather is expected to shift east on Sunday, giving the central United States a break. While there’s a chance for rain along the East Coast, the heaviest rains are expected in the southeast from the Gulf Coast to the Southern Appalachians. In northern Kentucky, another inch or two of rain is still expected before the long stretch of bad weather finally clears, said Nate McGinnis, a meteorologist with the National Weather Service in Wilmington, Ohio. But the flood risk is not as high as it was on Friday and Saturday.

Still, places across the storm’s path, including Louisville, Ky., and Memphis, have prepared for conditions to worsen. In Tyrone, Ky., where both roads into town were flooded, residents were evacuating on Saturday by boat as the waters rose. Bud and Tammy Morgan, who were waiting with their pets for an evacuation raft, planned to spend the night at a hotel and hope their home would survive.

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Elsewhere, people were contending with damage that had already been done.

In Hopkinsville, Ky., in the southwest of the state near the Tennessee line, floodwaters had inundated some homes, leaving mud on the floors and overwhelming losses.

“Everything is just destroyed,” said Brittanie Bogard, a city councilwoman, who began to cry as she described what she encountered when visiting flooded homes.

Though the rain slowed overnight, allowing the level of the North Fork Little River to drop several feet, it was still raining on Saturday morning and the river was beginning to rise again.

Jerry Gilliam, the judge executive for Christian County, which includes Hopkinsville, said emergency responders performed about 20 rescues from residences, and moved 40 dogs from a pet boarding service near the river. The county jail also sits right next to the river.

“We were just inches away from evacuating around 550 inmates,” he said. “But fortunately last night, that’s when the water started receding.”

Many of the towns experiencing flooding are accustomed to high water and seemed to be taking the conditions in stride.

In Cairo, Ill., which sits at the confluence of the Mississippi and Ohio Rivers, pumps were placed around town and a levee gate was barricaded as a moderate, steady rain fell on Saturday morning. Some streets were flooded, but no property damage had been reported.

“Everybody within the city is working around the clock to make sure we keep the water out,” said Romello Orr, a member of the City Council and a restaurant owner in Cairo.

Outside Winchester, Ky., where the Kentucky River runs, Karl Crase said the lower-floor dining area at his restaurant, Hall’s on the River, was underwater on Saturday. He was hopeful that this flood would not be as bad as some in the past.

“We get the beauty of being on the river and the benefits and the uniqueness that comes with that. Then we become Hall’s in the River,” Mr. Crase said. “It’s part and parcel of the life we lead.”

Reporting was contributed by Mike Fitzgerald in Poplar Bluff, Mo., Carly Gist in Cairo, Ill., and Ginny Whitehouse in Winchester, Ky. Simon J. Levien, Yan Zhuang and Jonathan Wolfe also contributed reporting.