Translation:The rain started while the children were swimming.
But try this way and it's accepted : It started to rain while the children were swimming. But of course, as Sabine said "While" is on "swimming", not on the "rain"
That başladı sounds a lot like a başlıdır, like, exactly the same to me as if it had an r. is that normal? will it in real life too?
This sentence is in two parts, each part with its own verb at the end. Your sentence would be something like 'Yağmur başlarken çocuklar yüzdü.'