The main counter to split pushing is control of your waves. Unless the tower is low or inhibitor exposed they are unlikely to be able to "back door" it, they need a minion wave at the tower.
So the first thing you do is make sure all your minion waves are always pushed out to the center line of the map. If you don't do that then you have no time to do anything about the split pusher as he can stay out of sight until the wave gets close to a tower and then rush in and take it.
The next thing is to wait until you see the split pusher moving into your half of the map. Once that happens you then have three choices. Whichever you do you need to do it immediately though, if you wait before moving then it will be too late and he will reach that tower.
Block them. Send one person who can stop them (either by wave clearing under turret or ideally kill them or drive them off). That person should be careful though. If they die the split pusher will most likely take the turret so it's more important to stop them and stay alive yourself than it is to go out hunting the split pusher unless you are absolutely sure you can kill them. The remaining team will need to fight 4v4 on the rest of the map so choose the blocker as someone who is less useful in a team fight.
Take objectives. If they have one split pushing they only have 4 elsewhere. Take baron, dragon, towers of your own. Ideally rush down mid as 5 and drop tower after tower, dive their 4 and kill them then take the base while 1 person recalls to stop the split pusher getting any further. Just make sure the objectives you take are worth the ones you are giving up.
Kill them. With vision control two or three people can sweep in and catch the split pusher and kill him. Just make sure you aren't giving up objectives elsewhere while multiple people chase the split pusher. If need be just scare him off, push the wave back out, then move back to other objectives.
You can also combine these strategies. For example take drake as 5 people that they can't contest with 4, one teleports top to defend the tower, then four people move up to cut off his retreat, kill him. Now take baron 5v4.
Note how you are using the fact that one member of their team is off doing his own thing to take advantages everywhere on the map. It requires good team co-ordination and responses but can be very effective.
But it all comes back to control of those waves. Make him sit in lane for at least a while pushing out minions if he wants to do a split push and that gives you time to make a response.
