Because there's no way a green boy would defeat him
When Robb Roose marches down the King's Road, Tywin yet again refuses to use his superior numbers and instead of regrouping with Jaime, eh decides to fight him on even odds (or it would have been even odds had Robb been there with the cav). Up next Tywin continues his attempts at being defeated in detail by placing himself at the Crossroads where either a Northmen or Vale force could easily defeat him, both with good reasons to join. Had Jaime lost Tywin's invasion would have been dead on it's tracks. There was no way Edmure had anything under 10k, probably more equal to Jaime's host. Instead of Tywin marshalling his fast armies and using his numbers to smash Edmure there, he sent only Jaime with 15k men, thus gambling the outcome of the battle. Case in point, the battle of the Golden Tooth was incredibly dangerous. Any force, be it the Riverlands, the North or the Vale could win even if they were in numerical inferiority by attacking one army and then the other.
First he divides his armies with no way for the two to reinforce each other, thus badly making himself vulnerable to defeat in detail. So Tywin might be considered a fearsome military ruler, but if one looks at his campaign of conquering the Riverlands, you'll find mistakes not even a noob Total War player would do.
