The worse than usual seasonal floods have been caused by melting snow.
Across the border in Russia, an oil refinery in the city of Orsk, 1,800km southeast of Moscow, has stopped operations because of the floods.
The Kremlin has warned that some regions of Siberia could also be affected as the flooding spreads.
Spokesman Dmitry Peskov said President Putin had ordered the governors of the Siberian regions of Tyumen and Kurgan to prepare for “expected sharp rise in water levels” and “inevitable” floods.
Russian authorities said on Saturday that they had evacuated almost 4,500 people in Orenburg region in the area around Orsk.
During a visit to Orsk on Sunday, Russian Emergency Situations Minister Alexander Kurenkov, warned that “a critical situation” had developed after a dam was breached on Friday.
Russia’s weather monitor has warned that water levels on the Ural river in the city of Orenburg will reach dangerous levels over the next three days.
Regional authorities said they expected the “peak” of the flood on Tuesday and for the situation to stabilise after 20 April.
Meanwhile in Kazakhstan, the emergency ministry said on Sunday that some 12,000 people were being housed in temporary shelters.
It said rescuers also managed to move about 60,000 farm animals to safe areas.
President Kassym-Jomart Tokayev said on Saturday that it was the worst natural disaster to hit Kazakhstan for 80 years.
In a televised address to the nation, Mr Tokayev said the government had ramped up efforts to mitigate the impacts of the floods and that all necessary assistance would be provided.