Nepal’s government is led by a 36-year-old former engineer turned rapper, Balendra Shah, who swept to power just months ago on the back of youth anger over an economy ravaged by corruption and mismanagement.

Now, he and his fledgling government face a herculean test after devastating floods ravaged his impoverished country. A majority of Mr. Shah’s 17 ministers are under 40, and they were still finding their way around governance before the disaster struck.

Five days after a deluge caused by a glacier breaking in the Himalayas flooded northern Nepal and parts of Tibet, the number of missing in Nepal jumped Monday to more than 4,700 and the death toll rose past 900 as the scale of the disaster became more apparent.

Mr. Shah has not spoken publicly since the floods. Rather, he has visited places affected by the disaster, wearing his trademark double-breasted suit, black shades and white sneakers. He has led meetings with his ministers, who have also been in the field overseeing emergency efforts.

“There can be no greater pain than losing loved ones, homes and livelihoods,” Mr. Shah, who prefers communicating on social media to taking questions from reporters, said in a statement on Facebook on Sunday. He added, “Without making excuses, we have mobilized the entire state machinery on a war footing and are working around the clock to save lives and provide relief to every citizen in the affected areas.”

The task is daunting. Entire towns have been wiped out, and tens of miles of infrastructure are devastated. Rescue operations at tunnels of hydropower projects have been slow, with the military struggling to drill openings through mounds of debris.

When Mr. Shah looks up from the immediate work of emergency response, he is likely to find a country with the deeper economic woes that inevitably follow natural disasters.

That is bad news for Nepal, which was already one of the world’s poorest nations and struggled to meet its basic needs. But analysts say it is even more demoralizing for a government that was grappling with how to deliver on the populist promises that helped it defeat the old guard.

Mr. Shah’s strength has been in a populist wave he rode by responding to demands from a large disillusioned youth population forced to seek jobs abroad. Even before the floods, analysts said, he began to realize the honeymoon period was over. His grip over the social media narrative was waning as he faced the realities of insufficient funds for his agenda and an entrenched bureaucracy.

“This government’s focus has been ‘Let’s build this economy, let’s create jobs, let us create small and medium enterprises, let’s create start-ups’” said Lila Pieters Yahia, the top United Nations official in the country. “For this government, who is putting the pillars, the foundation of a new economic model, having this hitting you now? You can imagine.”

Mr. Shah’s rise has been swift since he entered politics five years ago to run for mayor of Kathmandu, the country’s capital. His victory was an initial indication of how fed up young people were with politics as usual, as the country’s transition from monarchy and civil war had given way to misrule by entrenched dynastic parties swapping power.

The anger spilled onto the streets last year in protests started by Nepal’s Gen Z that toppled the previous government. After national elections this year, a large number of young and first-term lawmakers entered Parliament and Mr. Shah became prime minister, bringing many of the Gen Z protest leaders into his fold.

The work of changing governance has not been easy.

Nepal’s disaster response agency, a key institution, has long been caught in a tussle for power between government departments and has lacked the funding and expertise to plan vigilantly, analysts and former officials say.

Ajaya Dixit, the executive director of the Institute for Social and Environmental Transition, said Mr. Shah’s young team appeared to be doing its best in the face of a “colossal” disaster. But he said the problem was much deeper, with countries in the Himalayas ill prepared despite suffering nearly a dozen catastrophes over the past four decades.

“What have the states learned from these episodic events?” he said. “The learning seems very limited.”

Critics have accused Mr. Shah of using people’s dissatisfaction with the old ways of governance to centralize power in a small team of largely inexperienced young operators. That has created bottlenecks that have been apparent in some areas of the disaster response, where local administrations feel helpless because the central authorities have taken over command, analysts and former officials say.

The dissatisfaction around centralized governance had begun manifesting in protests even before the floods. Mr. Shah’s team members also appeared at odds with each other over finding an economic policy balanced between fulfilling his populist welfare promises and taking the austerity measures needed to put the economy on a path to growth.

“Changing a few people at the top cannot instantly transform institutions and a system that has evolved over decades,” said Uddbab Pyakurel, a professor of political sociology at Kathmandu University.