MIAMI—Jeb Bush, heir to one of American’s most successful political dynasties, sought to jump-start his presidential campaign Monday as a scrappy political outsider after failing in recent months to emerge as the undisputed front-runner for the 2016 Republican nomination.

“Not a one of us deserves the job by right of resume, party, seniority, family or family narrative,” said Mr. Bush, kin to two American presidents and a former two-term Florida governor. “It’s nobody’s turn. It’s everybody’s test, and it’s wide open.”

In the six months he has been building his campaign, Mr. Bush has dropped in the national polls and finds himself unable so far to break from a crowded pack of GOP candidates, even though he is expected to raise the most money and has nabbed name-brand political talent.

Mr. Bush is reviving the logo he used for three gubernatorial campaigns—Jeb!—to emphasize that he is running not as a political scion but as his own man. Casting himself as a can-do chief executive, Mr. Bush drew his strongest contrasts to date with GOP rivals serving on Capitol Hill, including his former protégé, Florida Sen. Marco Rubio.

“Executive experience is another term for preparation, and there is no substitute for that,” Mr. Bush, who appeared without jacket or tie, told a cheering crowd his staff estimated at 3,000 people. “We are not going to clean up the mess in Washington by electing the people who either helped create it or have proven incapable of fixing it.”