Consider British Airways as an example. Their website provides a list of car seats requirements and one of them is:
must be designed to be secured by means of a normal aircraft single lap strap ...
Therefore, it looks like you need to buy a car seat which meets the requirements or to check rules of your airline (if it is not British Airways).
Please note, that it is allowed to have an infant on person's lap and some airlines (e.g. EU) provide special belts in this case, however, it is not clear whether such belts or seating on lap are safe.
According to "Study on Child Restraint Systems" (2008) page 36 (43 in pdf) such belts are banned in the US (at least in 2008):
The supplementary loop belt (belly belt) is not permitted.
but seating on someone's lap is allowed (the information is from 2008):
The child is seated on the lap of an adult without using any restraint
However, the same study provides examples of incidents (page 29 or 33 in pdf) caused by turbulence, one of which is:
An unfastened infant of seven weeks suffered most serious head
injuries since its mother had not been able to hold it.
Finally, according to the Federal Aviation Administration (the national aviation authority of the United States):
... the safest place for your child on an airplane is in a
government-approved child safety restraint system (CRS) or device, not
on your lap ...
Thus, it looks like it is safer to buy a special seat and (according to FFA):
make sure your CRS is government approved and has "This restraint is
certified for use in motor vehicles and aircraft" printed on it.