A lot of the problems the Republican party is facing today are due to it having been captured by its extremist wing. Democrats wanting to push the party further left (or people who are so far left they can't stand the Democratic party or its nominal leadership) would do well to remember that.
Many of those folks won't think the Republicans are facing any problems, since they nominally control so much of the federal government, so here are some of the problems Republicans are dealing with:
Their President is incredibly unpopular, and his party's candidates are avoiding him in the midterms.
Mike Johnson, the Speaker of the House, only got the job because most Republican congressmen saw what had happened to their previous Speakers and didn't want it.
Lower-level courts are ruling against the President and refusing to extend the usual courtesies to federal law enforcement because this Administration has been caught misleading the courts so often.
The unpopular gerrymanders the President insisted on look like they may've undermined his goals of Republican control in TX and other states, by narrowing slim conservative majorities in critical districts.
The Supreme Court's popularity (and thus authority) continues to decline.
Immigration enforcement is undermining the economy and deporting socially-conservative Christians who would normally grow and support the Republican base - or making the US so bad a place to live that refugees who have been driven from their homes are avoiding it.
Anti-authoritarian and anti-oligarch movements are gaining more recruits and energy, putting into question the millions wealthy US individuals have spent trying to control the narrative and federal policies. At the same time, those wealthy individuals are also seeing their businesses become less popular and less profitable due to Trump's grift, tariffs, and war in Iran.
Many socially-conservative Republicans are frustrated that the Trump Administration has refused to release the Epstein files and is focused more on personal vendettas than on prosecuting criminals.
The quality of Republican candidates is shrinking (e.g. Ken Paxton in TX and Darlene Graham in SC) and that's making it hard for them to turn out voters.
Podcasts and news outlets which are traditionally conservative are backing away from the President and from his allies (e.g. Wall Street Journal)
Promises of help from the Republican party have been undermined due to their failure to pass legislation which addresses the concerns of the majority of Americans, and due to Congressional Republicans & the White House publicly backing now-unpopular measures like ICE expansion and the One Big Beautiful Bill Act.
Left-wing liberals should continue to push for their views, and should absolutely continue to push back against this Administration, its apologists, and its defenders. But they also need to understand that all the Republican problems described above were self-inflicted and could have been avoided if the Republican party had put more authority in the hands of _moderates_. A Democratic party which builds the habit of denigrating and disenfranchising moderates for short-term base-consolidation risks undermining itself with the same problems.