Now I’m trying to wrap my head around why Tywin decided, of all things, to make his battle-weary men march on foot day and night to reach the Ruby Ford. Setting aside the wagon idea, there’s the Green Fork itself for water transport (and he would later do the same with the Blackwater, so it’s certainly feasible). Or he could learn from Robb Stark and split his cavalry and foot. Is he just that mediocre of a general?

1. Tywin was trying to get to Riverrun in time to save the army there. 

2. If Robb defeated the army at Riverrun, Tywin desperately needed to get across the Trident before he was trapped.

3. Water transport requires large numbers of boats to be ready ahead of time. Mace Tyrell had organized a large number of barges that were ready to go at Tumbler’s Falls, but without that, you can’t really move tens of thousands of men, thousands of horses, and tons of supplies by river.