We consider the fundamental question of which evolutionary histories can potentially be reconstructed from sufficiently long DNA sequences, by studying the identifiability of phylogenetic networks from data generated under Markov models of DNA evolution. This topic has previously been studied for phylogenetic trees and for phylogenetic networks that are level-1, which means that reticulate evolutionary events were restricted to be independent in the sense that the corresponding cycles in the network are non-overlapping. In this paper, we study the identifiability of phylogenetic networks from DNA sequence data under Markov models of DNA evolution for more general classes of networks that may contain pairs of tangled reticulations. Our main result is generic identifiability, under the Jukes-Cantor model, of binary semi-directed level-2 phylogenetic networks that satisfy two additional conditions called triangle-free and strongly tree-child. We also consider level-1 networks and show stronger identifiability results for this class than what was known previously. In particular, we show that the number of reticulations in a level-1 network is identifiable under the Jukes-Cantor model. Moreover, we prove general identifiability results that do not restrict the network level at all and hold for the Jukes-Cantor as well as for the Kimura-2-Parameter model. We show that any two binary semi-directed phylogenetic networks are distinguishable if they do not display exactly the same 4-leaf subtrees, called quartets. This has direct consequences regarding the blobs of a network, which are its reticulated components. We show that the tree-of-blobs of a network, the global branching structure of the network, is always identifiable, as well as the circular ordering of the subnetworks around each blob, for networks in which edges do not cross and taxa are on the outside.