Claim #1: The internet is changing our brains (e.g., see here). This is a tautology, as everything we experience changes our brains. Even no activity should change the strength of our synaptic connections! Now, the typical connotation is that these brain changes are for the worse, as the assumption is that deviations from normal biology are bad. But from what I can tell, we just don't know enough about systems neuroscience to correctly evaluate the effects of web-induced brain changes. So let's stick to psychology and behavior, which brings us to...
Claim #2: Overall, the internet decreases our attention span and makes us less likely to engage in contemplative deep-reading and thinking (e.g., see here). Carr and Lehrer primarily extrapolate from controlled psyc studies to address this claim, but those do not longitudinally track the same subjects. Instead, Cowen seems more on track in focusing on the "market data." Unfortunately, longitudinal book reading stats are very difficult to come by, although see here for one aggregation of survey stats. Book revenues did not change much from 2002 to 2009 (see here), but that is confounded by pop growth, inflation, and the shrewdness of Amazon. So, we need better data. Another strategy is to look at previous tech advances and how see people's contemplative deep thinking has changed. I would say, not much. For example, introspection habits haven't don't differ drastically between the 150 CE Marcus Aurelius and the internet-era's own Katja Grace.
Claim #3: Developmentally, if one is not forced to focus for long periods of time often during adolescence, one will be less able to focus as an adult (e.g., see here). It is certainly true that teens will on average be more impulsive than adults, which may be because the amygdala and nucleus accumbens develop more quickly than the prefrontal cortex. So, it makes sense that internet multitasking will be particularly tempting to teens. Indeed, teens may lose time that could have been spent studying. But school is largely zero-sum anyways. Plus, once adolescents become adults their focus should improve. This is a very controversial claim though, as people differ both on their beliefs about focus and on their values with respect to paternalism.
Claim #4: The gains we reap from more immediate access to info and more efficient reading are not worth the costs of habitually skimming and a reduced willingness to commit to valuable but demanding texts (e.g., see here). I mostly agree with Steven Pinker's assessment of this claim as bullshit, because the internet is the best / only way to keep up with the exponentially accelerating increase in knowledge. But he also says that deep reflection, thorough research, and rigorous reasoning "must [!] be acquired in universities," to which I call bullshit. The existence of autodidacts and other individuals self-educated primarily on the internet proves the non-necessity of universities.
Full Disclosure: By the end of writing this post, I have 24 tabs open in my Firefox browser. Namaste, cabrónes.