There were scythed chariots as well in ancient times. One tactic that was used to counter them was to move to create gaps in your line and the chariot would move through the gaps then the riders would be pulled from the chariot and killed.

There was one recorded instance by Xenophon (I believe) where a guy didn't get out of the way and just dove on the ground and the chariot went right over him without causing any harm.

The scythed chariot was mostly just a specialty "trick" weapon.

With other armies that used chariots more frequently and as a larger percentage of their army the utility of the warmachine declines and disappears as the warriors are able to actually ride on horseback.

There were a couple of styles of using chariots, people could fight from them or they could use them as "battle taxis" to take warriors to the fight. From there they would disembark and fight on foot with the chariot waiting behind for a quick retreat if it proved necessary.

In those cases the tactic was just to beat the warrior on foot and not let him get away, of course.

In the former case in places like the middle east larger shields were developed that could be hidden behind while infantry archers fired volleys over the shield bearers at the chariots as they swerved into range.

So the Persian spara is an example of this. It is a wicker shield that is not so good at stopping an infantry man thrusting with a big spear but is very good at stopping arrows.

The tactic remains viable against cavalry too, of course.

Another tactic used by the Assyrians was to use slingers who fired indirectly. The Assyrians were one of the few armies in history to have formed, heavy slingers.