The improvised explosive devices were set off Saturday and Sunday around the city of Jalalabad, capital of the eastern province of Nangahar and known as a stronghold for the Islamic State-Khorasan (ISIS-K). Though they are both Islamist groups, ISIS-K opposes the Taliban, which it accuses of not being extreme enough.
The Islamic State group's Amaq News Agency said on its Telegram channel that six attacks Saturday and Sunday killed or injured over 35 Taliban members, Reuters reported.
Bilal Karimi, a Taliban spokesman, said a bombing Sunday in Jalalabad targeted a Taliban vehicle, killing one child and injuring two people, among them a Taliban member.
"We have started investigations into the incident to reach the culprits," Karimi said.
But three residents, speaking on the condition of anonymity because they feared angering the Taliban, told The Washington Post that at least three civilians were killed, including a child, and scores more injured in the attack. Residents said the blast also knocked down a major power line, though it was restored later in the day.
The violence followed a series of explosions Saturday. One resident said four people were killed and 22 injured in five incidents in the city.
Karimi said that only "minor blasts" took place Saturday and "a number of casualties were reported," without providing specifics.
ISIS-K previously claimed responsibility for an attack at the Kabul airport on Aug. 26, which killed some 170 Afghan civilians and 13 U.S. service members at the already chaotic end of the U.S. withdrawal of troops after two decades in Afghanistan.
The Taliban rapidly regained control of the country amid the hasty exit of U.S. forces last month. The extremist group has since faced pockets of public opposition, including a resistance movement in Panjshir province and street protests by women who oppose the Taliban's brutal gender-based restrictions.
At the same time, the political vacuum left by the swift fall of Afghanistan's Western-backed government could further embolden ISIS-K, analysts have warned. The Taliban and U.S. forces previously found common ground fighting the Islamic State affiliate.
The Taliban leadership has said the group has softened its stances since it last ruled Afghanistan, from 1996 to 2001, when it became an international pariah for policies such as banning education for girls. Over its last month in power, however, the Taliban has formed an all-male government, imposed dress codes and segregated women at universities, and told female employees to stay home. On Friday, the group ordered high schools for male students to reopen - but made no mention of schools for female students.