Health ministry spokesman Wahid Majroh said at least 40 people were killed and 140 wounded in the blast.
The Defense Secretary had already left the airport in the Afghan capital.
The bomber struck as Shiite Muslims gathered for a major religious festival.
More than 80 were killed in the Islamic State attack on protesters from the mainly Shi'ite Hazara minority group.
Tolo TV was attacked for "promoting obscenity, irreligiousness, foreign culture and nudity," the Taliban said in a statement.
At least four people have been killed and dozens have been injured in the attack.
The embassy urged extreme caution after receiving reports of an attack within 48 hours
Attack on NATO convoy leaves 12 dead in Afghanistan capital.
The Tarakhil Power Plant provides little electricity to the Kabul grid.
The explosion is the latest in a series of suicide attacks in Kabul following a change in Taliban leadership.
The bombings were the first in Kabul since Mullah Akhtar Mansour took charge of the Taliban last week.
The attack happened hours after a truck bomb detonated near the city center, killing 15.
Truck bomb kills at least six people in Logar province, south of Kabul.
Police have arrived at the scene in an upscale area of the capital.
Hundreds take to the streets of Kabul to demand justice after woman killed over false allegations.
The blast at the school killed at least one person, and occurred just six hours after the first bomb
The attack is the latest in a wave of bombings to hit Kabul as foreign troops prepare to withdraw