。
She was very strict with her child.
Literal
She [topic-は] her own child [toward-に] very strict was.
厳しい ('strict, severe, harsh') with に marks the target of strictness — '(she is) strict toward X.' This is the standard frame for adjective-of-attitude + に + target. 大変 ('very, greatly') here functions as an adverb intensifier in formal/written style — quite a bit more elevated than とても in casual speech. The plain past adjective 厳しかった describes an ongoing past state.