Images of the explosions show sheared metal and extensive damage to buildings.
China's new military base is just a few miles from the only permanent U.S. outpost in Africa.
Around 100 U.S. troops ended a six-year mission earlier in 2017 after failing to capture Kony, whose group has co-opted tens of thousands of children into conflict.
President Trump gave U.S. forces in Somalia more leeway to strike the Al-Qaeda affiliate earlier this year.
Obama has tried to turn the U.S.-Africa relationship from dependency to partnership.
Al-Shabab says its fighters killed five soldiers in the clashes, a claim Nairobi denies.