Children among those killed in separate unclaimed bombings in Kapisa and Kandahar provinces.
One of the explosions took place in Dar-e-Tapa village in the Nijrab district of Kapisa province, in an area where children were walking, provincial police chief Mohammad Razaq Yaqoubi told Al Jazeera.
The roadside mine killed two children and wounded another two.
"The wounded children were taken to hospital and now they are in stable condition," Yaqoubi said.
In a separate incident, one civilian was killed and 10 people wounded, including three policemen, in a double bombing in Kandahar city, capital of the synonymous province.
There was no immediate claim of responsibility for either bombing.
Afghan army advances in Kunduz
Separately, the Afghan military on Tuesday regained control of a district in northern Kunduz province that had fallen to Taliban fighters earlier this month as part of their so-called spring offensive, security officials said.
The army’s advance came after Afghan forces launched a major operation against Taliban positions to retake the Qala-e-Zal district.
"The governor’s building, police headquarters and several key areas are cleared of terrorists, but the operation is still ongoing in other insecure areas of the district," the interior ministry said in a statement.
Speaking to AFP news agency, Assadullah Sadat, a member of the Kunduz provincial council, said: "Around 2,000 families have been displaced with most relocating in Kunduz city. Some have fled to other nearby districts."
The Taliban had taken full control of the district on May 6.
The group’s so-called spring offensive normally marks the start of the fighting season, though the Taliban had continued to battle government forces through this past winter. An attack on a military base in the nearby city of Mazar-i-Sharif last month killed at least 135 security forces.
Kunduz is among the most violence-wracked provinces in northern Afghanistan. Although the city centre itself is in government hands, the Taliban control most of the surrounding districts.
Security forces have been struggling to open the main highway into the city after it was blocked with mines and improvised roadside bombs.
Thousands of residents are reported to have fled their homes to avoid the fighting.
According to US estimates, the Afghan government controls only about 60 percent of the country, with the rest under Taliban control or contested by armed groups.