Here's a quick video showing two forward-facing car seats in a crash test. The seat closest to the camera is installed properly (so that it moves less than an inch side-to-side when tugged). The seat behind it doesn't have a tightly locked seat belt and moves more than an inch - and therefore isn't able to do it's job.
So how can you make sure your car seat is installed properly?
Car seats achieve this 'solid' installation in a number of different ways...
Locking the seat belt
The most common way is to lock the seat belt. All modern seat belts will lock in an emergency (when the driver slams on the brakes or when you jerk on the seat belt). However, some seat belts have a second mode called child-restraint mode (also called ALR, auto-locking retractor). To engage child-restraint mode, simply pull all the belt slowly out of the retractor - when you feed the belt back in, you'll hear a ratcheting sound. This is the sound of child-restraint mode, woohoo! This means the belt will only get tighter (around your car seat) until you feed all the belt back into the retractor to reset the mechanism back to normal-mode. Check your manual, but the vast majority of seats can (and should, where available) be installed using the child-restraint mode of the vehicle's seat belt. In Singapore, very few taxis have dual-mode seat belts (some of the red TransCab Toyota Wish taxis do), but most of the Uber and Grab cars do have seat belts with child-restraint mode.
If your vehicle doesn't have seat belts with child-restraint mode, or if you catch taxis/Uber/Grab, you can easily lock the vehicle's seat belt using a locking clip. The standard locking clip is suitable for some car seats (typically US and Australian certified seats) installed permanently in private vehicles. The taxi-friendly locking clip can be used on almost any car seat in almost any car/taxi (it's also much easier to use).
As a general rule, all US-certified and Australian-certified car seats will require a locking clip or a seat belt with child-restraint mode. Check your restraint's manual to be sure.
While it might not sound particularly fail-safe, since European cars typically don't have seat belts with child-restraint mode, European car seats typically rely on friction between (i) the car seat and the vehicle upholstery and (ii) the vehicle seat belt and the child car seat to provide pre-crash locking. This is why most European seats have a very circuitous belt path, often passing the shoulder belt behind the back of a rear facing car seat. Some car seats will include a taxi-friendly locking clip - in which case we recommend you always use it.