I shall start by saying that this is a somewhat sensitive subject, as giving the wrong advice could have terrible consequences. If you are reading this as a way to make a decision about how to transport your child(ren) then I would encourage you to have a look at the data. Cars are really dangerous, and it seems there are things you can do to reduce the risk. Do not listen to me, I do not have children or any professional experience in road safety.
In trying to answer this as data question I started with a 25 minute youtube video that introduces most of the papers referenced below.
It is easy to find correlation between injury and death rates and child seat use. For example Elliott et al. 2006 found a relative risk for death of 0.72 (0.54-0.97 95% CI) associated with the use of a child restraint system in children of ages 2 to 6, and 0.79 (0.59-1.05) when excluding cases of serious misuse. This represents a 21% reduction in the chance of dying in an accident. Arbogast et al. 2009 find reductions of adjusted relative risk for injury in children of ages 4 to 8 of 0.55 (0.32–0.96), representing a 45% reduction in risk. This effect appears particularly pronounced in younger children, with Zaloshnja et al. 2007 finding a 81.8% lower chance of injury in children aged 2 to 3.
However this sort of study has been questioned in a number of ways. Probably the hardest to deal with is the question of how much of this correlation represents the use of child restraint systems causally resulting in lower rates of death and injury of the individual, and how much this represents the use of child restraint systems serving as a marker for wealth, the use of safer cars and/or more responsible driving behaviour. There are also questions about the validity of the data. The presence of child restraint systems is easier to prove than the use or not of a seat belt, so it is postulated that when insurance professionals ask parents of children who where unrestrained they may be incentivised to say they had a seat belt on. This would inflate the rate of harm associated with seat belt only use.
One study that attempted to control for these variables is Levitt 2005 who finds that child safety seats are no better than seat belts at reducing fatalities among children aged 2-6. Another is Anderson et al. 2009 that separates out booster seats and full seats with their own four point harness. This finds the estimates for chance of dying actually higher for booster seats than just seat belts, though none of the differences are statistically significant (Table 3, Panel II). Ma Et al. 2013 look specifically at booster seats compared to seat belts, and find significantly increased rates of some injuries, specifically "AIS > 0 injury to the neck and thorax".
It must be noted all these studies highlight the problem for incorrect installation of child safety seats as an unmodelled variable in the data, with the first two estimating this rate at 80% and >50% respectively. The importance of properly fitted seats is illustrated by Ciccone et al. 1997 who find at 25 mph "For the properly restrained dummies, the safety seats provided excellent protection from the crash forces. For the improperly restrained dummies, protection ranged from adequate to nonexistent, depending on the type of misuse." This could disguise a significant positive effect of properly installed seats.
On a similar vein Edgerton et al. looked at children between 20 and 40 lb in either a full child seat or booster seats and found odds ratios between 4.4 and 29 for various injury metrics. They interpreted this as the harm from switching from full child seats to booster seats too early.
There are few crash tests that directly compare seat belts to child seats. Kriston et al. (which is not peer reviewed, but published by the US DoT) compared a booster seat to seat belts with and without belt guides with a crash test dummy "aged" approximately 12 years, and find the booster seat 33% better (16%-50%). Talimian et al. 2023 do computer modelling and find a four point better that a three point, but their models do not look much like what I see sold as car seats.
Another point that is worth mentioning is the importance of position in the car on the chance of injury. As I have mentioned on mechanics.SE Arbogast et al. 2008 found "children seated in the center rear have a 43% lower risk of injury compared with children in a rear outboard position".