WASHINGTON — Democrats will soon get what they’ve long dreamed of: President Donald Trump will leave the White House. But even once President-elect Joe Biden takes office next month, many of their other hopes face extremely difficult roads to reality.

Biden will come to power with the smallest House majority in modern history, and will face a narrowly divided Senate where, no matter who leads the chamber, Republicans will still hold influence over what can advance.

So as Biden confronts twin health and economic crises, vows to rebuild the country, and aims to tackle climate change, any legislation with a chance to succeed will have to hold together the wide-ranging and sometimes unruly Democratic coalition — and likely need support from some Republicans.

Those dynamics weigh against some of the more ambitious and politically risky policies that some Democrats had envisioned pursuing if they had also decisively won the Senate. They might make even smaller legislation a slog. Much depends on the outcome of a pair of Senate runoff elections in Georgia next month, where Democrats could win control of the chamber — but with the slimmest possible margin.

“There’s no escaping the challenge we face even in a Senate that has a 50-50 split which can be broken by Vice President Kamala Harris,” Sen. Bob Casey (D., Pa.) said this month when he introduced a bill aiming to divert noncriminal 911 calls to social service agencies. “Even in those circumstances it’ll be difficult to pass legislation like this. Everything is difficult.”

He compared advancing bills to “pushing a heavy rock up a steep hill.”

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President Barack Obama and Trump each took office with stronger hands in Congress, and still saw key pieces of their agendas stifled.

Still, Democratic lawmakers are hoping that a sweeping coronavirus relief bill early next year — separate from the narrower one that came together this week — will provide a vehicle for many of their ideas and bring Republicans to the table. And they hope an oft-discussed national infrastructure program can gain bipartisan traction.

But even in a best-case scenario for Democrats, Biden will have to work with an evenly split Senate where Harris can cast tie-breaking votes. And that’s only if Democrats win both seats in Georgia.

“A lot of this is going to depend on the outcome of Georgia,” said Sen. Cory Booker (D., N.J.). ”If we lose Georgia, I don’t expect [Republican Senate Majority Leader] Mitch McConnell to change.”

Even if Democrats win, Senate rules require 60 out of 100 votes to advance nearly all legislation, so Biden would need to persuade at least 10 Republicans for any major push — while also holding together a Democratic caucus that ranges from moderate West Virginian Joe Manchin to progressive Vermonter Bernie Sanders.

Some Senate maneuvers can be used to pass bills with just 51 votes, as Republicans did early in Trump’s tenure. But they would have to fit certain criteria, and even then Democrats would have no margin for defections.

If Republicans hold the Senate, Democrats have an even tougher lift, since McConnell will control what gets a vote. Anything that can pass will need to be palatable to the GOP.

In the House, where Democrats will hold a slender majority after losing seats in November, several aides to Pennsylvania and New Jersey lawmakers worried that vulnerable members will be more cautious as some look over their shoulders for challengers in 2022. The party has already shown cracks as progressives and centrists clash over the best path forward.

“I don’t think we’ll see any dramatic policy initiatives enacted into law,” said Charlie Dent, a former Republican congressman from Allentown. “I hate to use that term, but small-ball ideas will certainly be in play.”

Two possible exceptions, Democrats hope, could be a bigger coronavirus relief bill and an infrastructure package.

With the pandemic still surging, millions out of work, and many businesses struggling, another relief measure could provide an opening for Democrats’ more ambitious plans — though many are sure to face opposition from Republicans, including Sen. Pat Toomey (R., Pa.), who has argued for more targeted spending.

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“What we have learned is it’s not enough to simply get back to normal. We need a new version of normal,” said Rep. Chrissy Houlahan (D., Pa.), of Chester County. “We have seen the reality that people were within one or two paychecks of economic devastation. We knew that was the case, but now we’ve seen it firsthand.”

Local members of both parties pointed to vaccine distribution as a shared, nonpartisan goal.

A relief bill could also become a magnet for all kinds of ideas as lawmakers try to attach their proposals to what Washington insiders call “a moving vehicle” — a rare proposal that seems destined for passage. That could mean appending plans for healthcare, the economy, infrastructure, and other issues where the pandemic exposed problems.

Rep. Dwight Evans (D., Pa.) hopes to see aid for “safety net” hospitals, including those in cities and rural areas, to prevent closures like that of Hahnemann Hospital in Philadelphia. He hopes an infrastructure bill would include aid to rebuild crumbling schools, one of his longtime priorities.

Rep. Madeleine Dean (D., Pa.), a former state lawmaker, argued that a relief package should include aid to state and local governments shouldering some of the burden of distributing vaccines. Houlahan hoped lawmakers might approve new money for training and education for frontline workers who want to advance their careers, “in the same way that we helped G.I.’s in the past with their service.”

Rep. Andy Kim (D., N.J.) said the plan should aid small businesses, like restaurants and bars, that have been hit hardest.

Some Democrats disputed the idea that Biden will face a rough road.

“I’m far more optimistic than those who are saying it’s going to be limited,” Evans said.

Dean argued that the split verdict in the election — Democrats winning the presidency while Republicans gained in the House and held several competitive Senate seats — should encourage cooperation.

“The election taught us, or commands us, to be more bipartisan,” Dean said.

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With such narrow splits in both the House and Senate, Rep. Brian Fitzpatrick (R., Pa.), said centrists who can form bipartisan coalitions will gain clout.

Fitzpatrick hopes to see a coronavirus assessment similar to the 9/11 Commission that examined failings after that tragedy. And like local Democrats, he embraced the idea of a major infrastructure program. “It will double as a stimulus bill,” Fitzpatrick said.

Several Democrats made the same case, arguing that it will create jobs and improve the economy by expanding and improving internet access, and making travel and transportation smoother.

“It’s one thing that philosophically will bring all of the various groups together,” Evans said.

Yet for years, members of both parties, including Trump, have talked up infrastructure, with little progress as the two sides split over how to pay for such a program, and how expensive it should be.

Democrats hope Biden, who was a senator for nearly 40 years and has worked with McConnell, can have more success collaborating with the GOP. Biden himself has argued that Republicans will come around to work with him on some issues, though many liberals dismiss the idea that he can expect collaboration with McConnell, who relished blocking Obama.

Biden will also have other tools of the presidency to advance his agenda, much as Obama and Trump did — once it became clear that Congress was going to seek to block them.

Staff writer Allison Steele contributed to this article.