The first explosion took place at a house rented by three Iranian men in the Klongton district, south of the city’s center, at about 2pm local time, police deputy commissioner Pisit Pisutsak said.
Two of the men fled the scene, and a third man, who was injured, then tried to take a taxi, Pisit said.
The taxi driver refused to pick up the man, so he threw a bomb, which rolled underneath the vehicle and exploded it, according to The Nation.
The man later attempted to throw a bomb at police outside the nearby Kasem Pitthaya School, but it slipped out of his hands and blew up.
The man lost both of his legs in the blast and was taken to the Chulalongkorn Hospital, the newspaper said.
The suspected bomber, identified by police as Saci Morabi, 50, was arrested. Police were searching for the two other suspects.
Police identified four people who were injured in the blast as Sinchai Boonsoongnern, Sangwal Horprasartthong, Juthatip Sajjadamrong and Apichart Khamlue.
They said Sinchai was the taxi driver but did not release any further details about the others.
Police later used high-pressure water cannon to defuse another device found inside the suspects’ house, according to AFP.
Thai Prime Minister Yingluck Shinawatra urged the public “not to panic” following the attacks.
“Let the police and intelligence agencies do their work, and the public must not panic because the perpetrator was detained,” she told reporters.
The Israeli embassy is within four blocks of the explosion, and a Jewish community centre is also close, according to Sky News.
The attacks come a day after an Israeli diplomat’s wife and three Indian nationals were injured in a car bomb in New Delhi. The attack, along with a separate car bomb plot against Israeli embassy staff in Georgia, put Jewish diplomats around the world on high alert.
Israeli Prime Minister Benjamin Netanyahu accused Iran of being behind the attacks, although Tehran immediately branded the accusation as “sheer lies.”
Last month, Thai police charged a Lebanese man, with alleged links to terror group Hizbollah, of planning an attack in Bangkok. The US had earlier issued a warning that foreign terrorists might be plotting attacks in Bangkok.