Three people were killed when three blasts struck the Afghan city of Jalalabad on Saturday.
At least one was aimed at a Taliban vehicle.
It was the first deadly terrorist attack in Afghanistan since the US withdrew from the country last month.
An official from the health department of Nangarhar province told AFP that three people died and 18 were wounded.
Later on Saturday, two people were injured in Kabul when an explosive device attached to a car was detonated, Afghan news service Etilaatroz reported.
“Sticky bombs” – magnetic explosive devices attached to cars – have been a common method of assassination in Kabul. But the tactic was previously used by terrorists against Afghans working with the internationally backed government, which fell on August 15.
The roadside bombing in Jalalabad was a more common tactic, locals said.
The Taliban stormed to power in mid-August, ousting the internationally recognised government and promising to restore security to the violence-wracked country.
But they have struggled to contain their rivalry with the Afghan ISIS affiliate, IS-Khorasan Province.
“In one attack, a Taliban vehicle patrolling in Jalalabad was targeted,” a Taliban official who asked not to be named told AFP.
“Women and children were among the injured,” he said.
Pictures taken at the site of the blast showed a green pick-up truck bearing a white Taliban flag, surrounded by debris, with armed fighters looking on.
Jalalabad is the capital of Nangarhar, the heartland of the Afghan ISIS affiliate.
Kabul airport attack
IS-K has been active in Afghanistan since about 2015.
On August 26, a bombing outside the gates of Kabul airport killed at least 170 Afghan civilians and 13 US soldiers amid a mass evacuation. The attack was attributed to IS-K.
That bombing led the US to launch a drone strike on a vehicle believed to belong to IS-K.
But the US said on Saturday that its military had been mistaken and that as many as 10 civilians, including seven children, had been killed.
Since the last American soldier left on August 30, Afghanistan has been free of bombs, air strikes and widespread fighting.
The attacks on Saturday suggest the current lull in violence may be only temporary.