Yes most times a car will still run even if the turbo goes out however although the car might still run with the turbo not working the engine will run poorly and you'll lose engine power.
So you could only drive the car with the turbo out for short distances until you can get the turbo fixed.
When a turbo on a car goes out it's usually due to bad seals or bad bearings and you'll notice a loss in engine power, poor acceleration and possible engine stalling.
It's also good to note that even when a turbo goes out it will seldom cause any engine damage but you should not drive too long with a bad or failing turbo.
You certainly won't have a good running car with a bad or failing turbo and your fuel consumption will increase as a result.