The phrase "became afraid that" is correct and usable in written English.
You can use it to describe a situation in which someone suddenly has a feeling of fear in response to something. Example: When the storm began to pick up, Jane became afraid that her house would not withstand the high winds.
There was a great deal of smoke; we even became afraid that we might choke to death".
They became afraid that people were starting to support his thinking and that this would cause trouble for them.
At the end of October, an officer with the Pacific Command who was providing Clovis and the others with support became afraid that he had been covertly slipped LSD, and was hospitalized.
The defendant became afraid that he was 'trapped,' that he would be discovered, and that someone would have a cell phone and he would be 'busted.' When the group was about 15 to 20 feet away, the defendant jumped up from his position.
Others have argued that Sibelius's exalted status as a national hero effectively silenced him; he became afraid that any further major work would not live up to the expectations of the adoring nation.
Many voters, considering the liberals part of the bedrock of postwar German politics, become afraid that the FDP gets ejected and "tactically" lend their second votes to the FDP so that it stays in.
But occasionally, people do queue up outside the bank - or empty their accounts electronically - if they suddenly become afraid that the money in their bank account is no longer safe.
When valuations are increasing exponentially, excitement is building to a fever pitch, and companies are sprouting seemingly overnight — it's almost like everyone becomes afraid that success won't last.
And those who have counted on us for so long may become afraid that after all the changes we are going through, we won't love them as devotedly as before.
