While Peter Turchin has overstated the case, it is most likely that this was usually true. Infant mortality rates were very high until they started to decline in the early decades of the 20th century (especially in Europe and the US). To this can be added the generally high death rate from lack of effective medical care and frequent epidemics. These factors meant that the birth rate had to be very high in order for it to exceed the overall death rate.
For ancient Rome, it is impossible to be certain but there is some fairly strong evidence that Turchin's statement can be fairly applied to the late Republic and early Empire (pdf).
There is considerable evidence to show that Roman society in the late
Republic and early Empire was afflicted by a low birthrate. Augustus
in 18 B.C. found it necessary to pass the lex Iulia de maritandis
ordinibus in the hope of raising the birth-rate by penalizing the
unmarried and the childless. In 9 A.D. he attempted to supplement this
law with the lex Papia Poppaea. The very existence of this legislation
indicates that the problem of childlessness was widespread and
long-Iasting, a view which is further supported by references to this
subject in Latin literature).
The causes of this low birth rate are disputed by scholars with reasons ranging from too many hot baths adversely affecting male fertility to the use of contraception and abortion. What other evidence we have comes from funerary monuments, and sources commenting on factors such as epidemics, seasonal deaths and birth control. None of these, though, allow us produce any firm statistics and for other periods in Rome's history we have much less information.
In the Princeton/Stanford Working Paper in Classics Disease and death in the ancient city of Rome (pdf), Walter Scheidel states that the demographic 'balance sheet' is "profoundly unclear", adding that
A very rough guesstimate for the early modern city of London
envisioning an annual excess of deaths over births equivalent to 1% of
the total size of the population has repeatedly been acknowledged by
Roman historians, yet its relevance is doubtful: it might equally well
be too pessimistic (because imperial Rome enjoyed better
infrastructure and welfare provisions) or too optimistic (because
London was free from falciparian malaria).
Thus, as Scheidel notes:
It is important to remember that not all premodern metropolises were equally deadly,...
The reference to early modern London leads to more concrete evidence outlined in The Population History of England 1541-1871 by E. A. Wrigley, R. S. Schofield and Roger Schofield. By looking at baptism and burial records for London from 1550 to 1824, the authors tentatively conclude that
it would appear that the latter [burials] regularly exceeded the former [baptisms] until 1802
Although the pattern was more pronounced in London than elsewhere, they also found that deficits in urban parishes and market towns were more common than surpluses. Nonetheless, there were urban areas which did have surpluses of births over deaths for several decades at a time.