On Tuesday Mr Bush accused the Obama administration of a “premature withdrawal” of US forces from Iraq in 2011, with “grievous” costs.
Ms Clinton replied by saying it was Mr Bush’s brother George W Bush who, as president, negotiated a US withdrawal.
The US-led war in 2003 has been followed by years of turmoil.
Mr Bush called the withdrawal of US forces in 2011 a “fatal error”, destabilising the nation and setting the stage for the rise of Islamic State militants.
“So eager to be the history-makers, they failed to be the peacemakers,” Mr Bush said of Mr Obama and Mrs Clinton, who was Barack Obama’s secretary of state from 2009 to 2013.
“Rushing away from danger can be every bit as unwise as rushing into danger,” he went on.
Analysis: Anthony Zurcher, BBC North America reporter
When Jeb Bush blasted Hillary Clinton for “losing” the Iraq War earlier this week, it probably suited her just fine. The challenge for the Democratic front-runner since she first declared in April has been to generate enthusiasm from the party’s rank and file for her seemingly inevitable march to the Democratic nomination. The more she mixes it up with Republicans, however, the more her faithful are likely to close ranks behind one of their own.
That, at least, seems to be outcome for which the Clinton campaign is hoping. And so, at the Iowa State Fair the day after Mr Bush made his rounds among the fried food stands and carnival rides, Mrs Clinton loaded up and returned fire at the Republican.
A war of words with Mr Bush could end up being the best way for Mrs Clinton to move on from the controversy surrounding her use of a private email server while she was secretary of state and the recent surge of fellow Democratic candidate Bernie Sanders of Vermont in some polls.
On Saturday Ms Clinton responded by saying Mr Bush “should present the entire picture. [That]… includes the agreement George W Bush made with the Maliki government in Iraq that set the end of 2011 as the date to withdraw American troops.”
“I can only wonder whether he either did not know that or thought that other people would not be reminded of that,” she went on.
Earlier in the campaign Jeb Bush was ridiculed for struggling to say whether he would have approved the Iraq invasion “knowing what we know now”.
At first, he said he would, then he said he wouldn’t engage in “hypotheticals” and finally he announced he would not have.