In dynamical models of cortical networks, noisy inputs can be amplified into structured activity fluctuations by the recurrent connectivity, essentially through a combination of two distinct mechanisms. First, those patterns that self-reproduce by passage through the connectivity matrix W display large but slow fluctuations (dynamical slowing). Second, if W is nonnormal in the mathematical sense, it may hide a functionally feedforward network of strongly coupled activity patterns, allowing for transient amplification on fast time scales. The latter mechanism of nonnormal amplification has recently emerged in the neuroscience literature, though only in the context of a network with specific structure [Murphy and Miller (2009)], or networks explicitly designed to exhibit the phenomenon [Ganguli et al. (2008); Goldman (2009); Benayoun et al. (2010)]. It is not clear to what extent nonnormality affects the dynamics of more generic models of cortex. Here we investigate the tradeoff between nonnormal amplification and dynamical slowing in large random neuronal networks composed of excitatory and inhibitory neurons. Assuming linear stochastic dynamics, we derive an exact expression for the expected amount of purely nonnormal amplification. We find that nonnormality primarily gives rise to macroscopic fluctuations of the global population firing rate, which explains the positive mean pairwise correlation among network neurons. Amplification along more detailed spatial patterns is microscopic, however, and its total amount is very restricted if dynamical slowing needs to be kept low. Thus, in order to achieve strong transient amplification with little slowing, the connectivity must be structured, so that the synaptic strengths can afford larger values – a region where amplification in fact explodes – while self-reproducing patterns are discouraged. We discuss why this could be a desirable feature for sensory cortices that need to track fast-changing signals, and we give a plausible example of structure that favors nonnormal amplification.