How do you think things would have turned out differently if Robb had kept the Riverlords with him rather than giving them leave to scatter and try to defend their individual lands?

So part of the issue here is the strange problem with the Riverland’s military numbers (are they 20k strong? Are they 40k strong? Are they 11k strong?). 

(Btw, just to correct something that seems to be cropping up in the fandom: “it had been at Edmure’s insistence that Robb had given the river lords leave to depart after his crowning, each to defend his own lands” – not Robb’s idea)

To me, the main issue is that the dispersal of the Riverlords gives Robb fewer forces he can use to accomplish his objective, which is a big part of the reason why he decides to take his cavalry to the Westerlands where he can do more with less, rather than engaging directly with Tywin. 

Now, obviously attacking Tywin directly is a mistake – Harrenhal gives huge defensive multipliers that would make an assault a bloodbath, and it leaves Robb’s rear open for Stafford’s army (which was Tywin’s initial plan). However, if it were me in Robb’s place I would have had the Riverlords swear an oath that they will liberate the Riverlands collectively before any man returns home to his own. Then I would have sent the Riverlords and their 20k men (minus a solid garrison at Riverrun) sweeping west to east to push Tywin’s reavers back to Harrenhal, preventing further damage to the Riverlands. (I would have also demanded that Roose occupy the crossroads and the Ruby Ford to prevent raiding over the Ruby Ford.)

I would have then ordered them and Roose to cut Tywin off from supply, and prepare ambushes if he tries to break out to the Westerlands – in other words, rather than trying to stop his march, turn it to something more like the guerrilla campaigns waged against the greens during the Dance. In the meantime, I would have taken the additional 5,000 cavalry available to me (assuming that the Riverlands = 20,000 men and 25% are cavalry as per norm) to mount a more aggressive campaign in the Westerlands.

And with 11,000 fresh and victorious cavalry under my command, I would have attacked Tywin’s exhausted army and attempt to destroy Tywin outright.