Some of the determinants of fertility decline, such as female education (everywhere) and government population control initiatives (most famously China, India, and Iran) are widely accepted. Others, such as the effects of media and nebulous culture, are more controversial.
Evidence from Brazil suggests that yes, media does have an independent effect on fertility. This study looked at the effects of one particular type of media: Globo soap operas, access to which was rolled out differentially from 1970-1990 and which were incredibly popular. 
Brazil's fertility collapse 1960-present was extremely rapid. From a TFR of 6.3 in 1960 to 2.3 in 2000 to 1.66 in 2023. This without any active state-enforced population control measures and with abortion (and, for a time, even advertising contraception) illegal.
After controlling for many, many potential confounders (including all of the obvious ones, such as development, education, religion, etc), this study found that Globo access explains about 7% of Brazilian fertility. This is nontrivial; 2/3 the effect of increasing education.
Globo soap opera female characters were much more likely to be childless or have small families (1 or 2 kids) then Brazilians, less likely to marry, more likely to cheat and divorce, and more likely to see upward mobility. This is posited to change people's family expectations.
This is their model and a table containing the various observed variables they controlled for. They also controlled for the general decline in fertility across the country to ensure they were only measuring area-specific effects.
These effects are heterogenous: poorer and less educated (read: illiterate, since this is Brazil) women are more affected. This makes sense, since TV soap operas are smaller and less influential part of the media diet of intelligent people.
There is also age heterogeneity - it affects older women more then younger women. This also makes sense, as unlike, say, the US, Brazilian fertility decline is largely driven by stopping births earlier rather than starting later.
They use multiple falsification checks to test for robustness. In particular, their estimated effects do not begin until 1 year after Globo access to an area.
They also demonstrate that it appears to be the soap operas in particular, and not Globo or TV generally, by showing the effect can be predicted by the change people naming children after soap opera characters and is stronger among women more like that year's female characters.
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