Romer and her husband did find a substantial impact of exogenous tax cuts on output, as Keynesian theory would predict. The way they talked about is suggestive of a demand side effect because they talk about the impact of tax cuts as a percent of output on dollars of output. They say that their results are "suggestive of demand effects", but they are careful to note that they can't nail down whether this is a demand-side effect or a supply-side effect (which is presumably why they look at the 1920s and the 1930s to look specifically at marginal tax rates in the article Bob cites).
The nice thing about the paper that Romer cites that differs from the papers that Bob cites is that it's a review article of a wide range of studies on the supply side effects of marginal tax rate changes in the Journal of Economic Literature. These survey articles should always be your starting point for figuring out what to make of a question in economics.
I am not equipped to evaluate the two papers that Bob picks out to support his case and where they fit in the literature, but I found this passage in the JEL article illuminating:
"Second, while there is compelling U.S. evidence of strong behavioral responses to taxation at the upper end of the distribution around the main tax reform episodes since 1980, in all cases those responses fall in the first two tiers of the Slemrod (1990, 1995) hierarchy—timing and avoidance. In contrast, there is no compelling evidence to date of real economic responses to tax rates (the bottom tier in Slemrod’s hierarchy) at the top of the income distribution. In the narrow perspective where the tax system is given (and abstracting from fiscal and classical externalities), the type of behavioral response is irrelevant. However, in the broader perspective where changes in the tax system such as broadening the tax base, eliminating avoidance opportunities, or strengthening enforcement are possible options, the type of behavioral response becomes crucial. While such policy options may have little impact on real responses to tax rates (such as labor supply or saving behavior), they can have a major impact on responses to tax rates along the avoidance or evasion channels. In other words, if behavioral responses to taxation are large in the current tax system, the best policy response would not be to lower tax rates, but instead to broaden the tax base and eliminate avoidance opportunities to lower the size of behavioral responses. Those findings also highlight the importance of the fact that the ETI is not an immutable parameter, but can be influenced by government policies. For this reason, it is likely to vary across countries and within countries over time when non-rate aspects of tax systems change."The first bolded sentence highlights why a lot of economists have doubts about the supply side effects of marginal tax rate changes. Yes, as the Romers point out reducing taxes has an impact on output, but it does not seem to change labor supply - only the way people report income. This should not be particularly surprising - did anyone actively pick up more hours after the Bush tax cuts? Has anyone talked to the boss this week about reducing their hours in response to the jmp in payroll taxes? Some peoples' tax planning may change (that's the timing and avoidance point they raise), but I don't think this result that real economic responses are weak. The one exception is at the lower end of the income distribution with people responding to the EITC, but that's not really what people are arguing about. I think there's probably broader room for agreement about some supply side effects for people living in those sorts of conditions.
The second bolded statement is important too when we think about the estimates that Bob cites that cover OECD countries.
Generally speaking I don't think Romer was contradicting herself but I don't think she made that super clear in the New York Times article for people familiar with her tax research either.
When you review the articles Bob cites (I don't have the time to) keep this in mind from the JEL article:
"The empirical methods are most convincing in estimating the short-term response to tax rate changes, and in that case one must be careful to distinguish the response to anticipated versus unanticipated changes. Estimates of the elasticity of taxable income in the long run (i.e., exceeding a few years) are plagued by extremely difficult issues of identification, so difficult that we believe that there are no convincing estimates of the longrun elasticity of reported taxable income to changes in the marginal tax rate. Analysis of panel data does not seem likely to resolve the identification issues raised by trends in income inequality and mean reversion at the top and bottom ends of the income distribution. Repeated cross-section analysis based on longer time series is very useful to analyze changes in the share of income accruing to various income groups and to assess whether those changes are systematically related to changes in marginal tax rates. However, evidence from a single country is in general not enough to conclusively estimate behavioral elasticities because many factors besides tax rate shape the income distribution. Timeseries share analysis, coupled with compositional analysis, can be useful to detect large short-term changes due to behavioral responses."