Texas was bracing for Tropical Storm Beryl as it crossed the Gulf of Mexico early Saturday, and a hurricane watch was issued for the state’s southern coast.

Hours earlier, the storm made landfall in Mexico as a Category 2 hurricane. Damaging winds and storm surge battered the Mexican coast, leaving snapped trees and power outages in its wake.

Beryl, which then weakened to a tropical storm, was expected to become a hurricane before reaching the Texas coast as soon as late Sunday.

In the past week, the storm flattened islands and killed 12 people in Grenada, Jamaica and Venezuela.

The storm has left destruction in its wake.

Forecasters predicted that Beryl would hit Mexico twice. It crossed the Yucatán Peninsula on Friday, and then, after traversing the Gulf of Mexico over the weekend, it was expected to reach the coast of the northern state of Tamaulipas, where a hurricane watch was in effect.

Dan Patrick, the lieutenant governor of Texas, who is serving as acting governor while Gov. Greg Abbott travels abroad, issued a disaster declaration for 40 counties expected to be affected by the storm. The declaration enables state resources to assist in local preparation and recovery efforts.

Mr. Patrick said that emergency response teams were paying close attention to areas at greater risk, from Brownsville, Texas, to the Louisiana border.

“Everyone along the coast should be paying attention to this storm,” Mr. Patrick said. The Weather Service forecast “damaging hurricane-force winds and life-threatening storm surge” along portions of the lower and middle Texas coast late Sunday into Monday.

Michael Brennan, director of the National Hurricane Center in Miami, said Saturday that when the hurricane makes landfall in Texas it could bring storm surge of up to five feet above ground level off the Gulf of Mexico in Matagorda Bay, where Matagorda County officials issued a voluntary evacuation order late on Friday.

Further south, officials in the Rio Grande Valley and Corpus Christi were distributing thousands of sandbags to help people prepare for potential flooding.

In Mexico, no injuries, deaths or major flooding had been reported as of Friday evening, Laura Velázquez Alzúa, Mexico’s coordinator of civil protection, said at a news conference. But crews in Quintana Roo were working to restore power outages affecting 20 percent of the population.

The storm’s effects were felt unevenly in Tulum, a resort town. Some tourists strolled along the roads to observe the damage, mostly downed trees and utility poles. Nearby, about 300 low-income residents lined up at a mobile kitchen set up by the Mexican army.

“I didn’t buy any groceries or anything,” said María Cruz de la Cruz, referring to her lack of preparation for Beryl. “There is no money to buy things.”

Early Friday, Mexico’s meteorological agency said the storm had dumped six to 10 inches of rain in Campeche, Quintana Roo and Yucatán. It reported wind gusts as high 135 m.p.h. and a storm surge of more than 10 feet along the coasts of Quintana Roo and Yucatán.

Earlier this week, at least 12 people were killed as the storm lashed parts of Grenada, Venezuela, then Jamaica and the Cayman Islands.

Beryl made landfall on Monday in Grenada, where officials said about 98 percent of the buildings on Carriacou and Petite Martinique, home to 9,000 to 10,000 people in total, had been damaged or destroyed, including Carriacou’s main health facility.

Aid was just starting to arrive by boat on Carriacou on Friday, after rough seas and destroyed roads complicated response efforts.

Beryl killed three people and destroyed 400 homes in Venezuela, President Nicolás Maduro said at a news conference on Thursday. The country on Friday was recovering from the havoc wreaked by the downpour and gusts, including flooded towns, homes engulfed in landslides and damaged schools and bridges, he said at another news conference.

In Jamaica, the storm was the strongest to approach the island in over a decade. Over half of the customers of the country’s main power provider remained without electricity on Friday, the company said. The storm also ripped away a section of the jet bridge roof of Jamaica’s main airport.

This hurricane season is expected to be busy.

Forecasters have warned that the 2024 Atlantic hurricane season could be much more active than usual.

In late May, the National Oceanic and Atmospheric Administration predicted 17 to 25 named storms this year, an “above-normal” number and a prediction in line with more than a dozen forecasts earlier in the year from experts at universities, private companies and government agencies. Hurricane seasons produce 14 named storms, on average.

Johnny Diaz, John Yoon, John Keefe, Mike Ives, Kenton X. Chance, Jovan Johnson, Emiliano Rodríguez Mega, Daphne Ewing-Chow, Aimee Ortiz, Derrick Bryson Taylor, Julius Gittens, Remy Tumin, Jesus Jiménez, Orlando Mayorquín, Sharefil Gaillard, Derek M. Norman, Christine Hauser, Emily Schmall, Linda Straker, Yan Zhuang, Ricardo Hernández Ruiz, Edgar Sandoval and Claire Moses contributed reporting.