We study two closely related processes on the triangular lattice: frozen percolation, where connected components of occupied vertices freeze (they stop growing) as soon as they contain at least $N$ vertices, and forest fire processes, where connected components burn (they become entirely vacant) at rate $ζ> 0$. In this paper, we prove that when the density of occupied sites approaches the critical threshold for Bernoulli percolation, both processes display a striking phenomenon: the appearance of near-critical "avalanches". More specifically, we analyze the avalanches, all the way up to the natural characteristic scale of each model, which constitutes an important step toward understanding the self-organized critical behavior of such processes. For frozen percolation, we show in particular that the number of frozen clusters surrounding a given vertex is asymptotically equivalent to $(\log(96/5))^{-1} \log \log N$ as $N \to \infty$. A similar mechanism underlies forest fires, enabling us to obtain an analogous result for these processes, but with substantially more work: the number of burnt clusters is equivalent to $(\log(96/41))^{-1} \log \log (ζ^{-1})$ as $ζ\searrow 0$. Moreover, almost all of these clusters have a volume $ζ^{- 91/55 + o(1)}$. For forest fires, the percolation process with impurities introduced in arXiv:1810.08181 plays a crucial role in our proofs, and we extend the results in that paper, up to a positive density of impurities. In addition, we develop a novel exploration procedure to couple full-plane forest fires with processes in finite but large enough (compared to the characteristic scale) domains.