Democrats had hoped the party’s convention last week in Philadelphia would win over skeptical voters and ease concerns about Hillary Clinton’s trustworthiness, giving her a slight advantage in an unpredictable election year.

But after Donald J. Trump criticized the parents of a slain Muslim-American soldier, that cautious optimism morphed into a widespread belief that the race had fundamentally shifted in Mrs. Clinton’s favor.

“It’s a more permanent turning point,” said Senator Kirsten Gillibrand of New York.

Allies remain skittish and say that by many measures, Mrs. Clinton is a weak candidate with a muddled message who faces an electorate in which a majority of voters do not trust or like her.

But Mr. Trump’s inability to seize on his own party’s convention and emerge a more disciplined candidate has eased early concerns that he could appeal to a broader electorate in the fall.