Now, as Mr. Trump prepares to appear with Mr. DeSantis at a rally in Tampa Tuesday night, Mr. Putnam is facing a double-digit deficit in the polls and odds so long that even some of his admirers suggest he should stop spending money attacking his rival and begin pondering a comeback after the Trump era has passed.

Mr. Putnam’s collapse and Mr. DeSantis’s rise illustrate the extraordinary clout Mr. Trump now wields in his adopted party, a power so great that the president is effectively able to decide primaries with a single tweet.

In recent weeks, the president has leapt enthusiastically into contested nomination fights, doling out endorsements — sometimes several a day — as part of an immersion into primaries that has caught contestants, Republican officials and in some cases even his own staff by surprise. By taking sides in intraparty disputes in a way his predecessors studiously avoided, Mr. Trump has helped put his favored candidates over the top in Alabama, South Carolina and most recently Georgia, where he helped a hard-right Republican prevail over a more mainstream candidate in the runoff for governor.

And there may be more to come: Mr. Trump recently said he would campaign “six or seven days a week” this fall for vulnerable Republican candidates — though some of them may not want his help in states or districts where he is unpopular.

[Here’s what’s coming up next on the primary calendar.]

His intervention in Florida has irritated everyone from grass-roots activists to Republican governors, who worry that Mr. DeSantis’s close ties to the White House will make him a weaker general election candidate than Mr. Putnam. In an interview, Mr. DeSantis declined to name a single issue on which he disagreed with Mr. Trump and said he would enthusiastically welcome him to Florida this fall.