Wiktionary
stretch the truth
vb. (context idiomatic English) to exaggerate, often to the point where the truth is obscured or lost.
Usage examples of "stretch the truth".
True, he would have to stretch the truth a little in filling out the form, and a little falsification of signature was involved, but his motives were pure, and, thus, his conscience was clear.
The moment had come, Silk felt, to stretch the truth in a good cause.
They had no idea how far Giles Ducroy might go and they realized that the man would certainly stretch the truth in order to clear himself with the airport officials.