Overall, it doesn't seem to have been very common. In fact, in the 19th-early 20th century historians and archaeologists had started to believe it was entirely Biblical and Roman polemic about their neighbors, because that was the only evidence. Until 1921, the only evidence we had was from monotheistic Hebrews accusing pagan apostates and Canaanite neighbors, and Romans accusing the Carthaginian. Neither context is really a credential for historical evidence, given the animosity between the groups involved.

Then in 1921, archaeologists at Carthage uncovered the Tophet of Salammbo. The soil was full of burnt wood, from multiple sacrificial pyres, and bones. Mostly the bones were from animals, but mixed in were the bones of young children. Further research indicated that they were all healthy. This all but confirmed descriptions like this one from Diodorus Siculus:

There was in their city a bronze image of Cronus extending its hands, palms up and sloping toward the ground so that each of the children when placed thereon rolled down and fell into a sort of gaping pit filled with fire. (Bib. Hist. 20.14.6)

Over time, excavations have unveiled multiple levels of burials and urns containing the remains of sacrificed, generally healthy, children. Typically they were younger than five years old, by our estimates. However, there are also the borns of infants, stillbirths, and underdeveloped fetuses, indicating that the naturally dead from complicated births were sacrificed somewhat more often. The sacrificial site is estimated to have been more than acre in area by the Punic Wars, and even in its diminished state after that, the Carthaginians continued to sacrifice there up to Christianization. There's a pretty clear correlation between the number of children sacrificed and periods of crisis. The worse off Carthage was, the more children went to the gods to try and secure their favor.

The word "Tophet" meaning "a place of burning" and comes from the Biblical Hebrew description of such sacrifices. This use is somewhat confused by the use to also refer to any place of human cremation, even for dead adults. Sometimes this latter version is used by archaeologists, but the example at Carthage is meant in the former sense as described here:

And they have built the high places of Topheth, which is in the valley of the son of Hinnom, to burn their sons and their daughters in the fire; which I commanded not, neither came it into My mind. (Jeremiah 7:31)

Since 1921, other Tophet sites have been found in Carthage, as well as the city of Motya off the western coast of Sicily.

Meanwhile the Bible, it's brought up numerous times in numerous contexts. In Genesis, Abram is commanded to sacrifice his son Isaac. In Leviticus, the practice is banned for the Hebrews, throughout the books of history and the prophets, its condemned mutliple times as something that Hebrews who slip into Canaanite religion do, particularly in Jeremiah. It's even practiced by two Judahite kings: Jehoahaz and Manasseh. Unlike Carthage, no sacrificial tophets have been found in the region of Israel/Palestine.

Interestingly, this practice is not referenced by any other groups in the region to describe any other groups in the region. It only seems to be applied by the Romans to the Carthaginians (which we have good physical evidence for) and by a variety of Biblical authors for the Canaanites in and around their kingdoms (which we only have literary evidence for). There is neither any reference or evidence for it in the Bronze Age Canaanites, or their Iron Age descendants: the Phoenicians.

In Phoenician culture, it only appears in the western colony cities, so far as we can tell. The Bible mentions it so thoroughly and in such different books that it's hard to think that it's pure polemic, especially when the Canaanite's cultural cousins in the west were also practicing child sacrifice. On the other hand, what connection, if any, existed between the polytheists of Judah and the Carthaginians is entirely unknown to us.

It's possible, though entirely unproven, to suggest that Canaanites from the southern Levant joined their cultural relations from Tyre and settled in Carthage. It's possible to suggest that two traditions developed independently, though this seems unlikely because they are so similar. It's possible to suggest that this was an aspect of Phoenician culture that was just glossed over in our surviving sources, either because it was covered in works that are now lost or was treated as common knowledge. Really, who knows.

As an addendum: The Quran also, very briefly, mentions that some polytheist Arabs practiced child sacrifice, but does not describe it, and suggests that the deed itself will inherently bring destruction on those who practice it. For reference:

Even so, in the eyes of most of the pagans, their "partners" made alluring the slaughter of their children, in order to lead them to their own destruction, and cause confusion in their religion. If Allah had willed, they would not have done so: But leave alone them and their inventions. (6.137)

I bring this up only because the gods, stories, and underpinnings of religion for other Semitic-language speakers tended to be related, and it has been speculated that there may have been a connection between beliefs like those in Phoenicia, Syria, and Mesopotamia, and the poorly recorded ancient Arab religion. If that speculation is true, then it would represent some sort of precedent within the wider religious tradition for child sacrifice. As it stands, we know so little about Pre-Islamic Arab beliefs that it may be a pointless aside.