What do you think would have happened if Robb had agreed to march on Harrenhal with the full strength of the North & Riverlands right after the Battle of the Camps, as Lord Blackwood suggested? The combined Stark-Tully host outnumbered Tywin 2:1 but Tywin had Harrehal’s fortifications+ more heavy horse on his side(likely). Could Robb have besieged Harrenhal & starved Tywin out? Also, what if the Stark-Tully host made an all out attack on Casterly Rock, as Marq Piper adviced?

Assaults would have been extremely costly – castles offer extremely high defensive multipliers, and historically garrisons of a few hundred or even a few dozen men could easily hold off thousands of men. Moreover, while the combined force had a 2:1 advantage, the usual rule of thumb for battles in the open field is that the attacker usually should have a 3:1 advantage. 

Heavy horse is unlikely to be an issue. Firstly, you can’t really use cavalry in assaults unless you dismount them. Secondly, Tywin is highly unlikely to have more heavy cavalry: he had approximately 7,500 horse at the Battle of Green Fork, and Robb alone had 6,000 horse at the Whispering Woods. Even after the Riverlands forces are dispersed following the Battle of the Camps, Edmure pulls together 3,000 horse at the Battle of the Fords. So Robb would have had at least 9,000 to Tywin’s 7,500. 

The bigger issue would be supply – after Tywin’s initial rampage across the southern Riverlands, his reaving after the Green Fork, and the scorched earth tactics of the Riverlords, and camping his army for months on end, the area around the Godseye would have been stripped bare of food and other supplies. 

If they’d attacked Casterly Rock, Tywin would have marched west same as in OTL. If you mean Edmure wouldn’t be around to stop him crossing the Red Fork, then Tywin is too far away to make it back to King’s Landing, the capitol falls, now Tywin’s a rebel, and Tywin would have been facing an army twice his own size out on open ground.