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For a multigraph F , the k-subdivision of F is the graph obtained by replacing the edges of F with pairwise internally vertex-disjoint paths of length k + 1. Conlon and Lee conjectured that if k is even, then the (k − 1)-subdivision of any multigraph has extremal number O(n 1+ 1 k ), and moreover, that for any simple graph F there exists ε > 0 such that the (k − 1)-subdivision of F has extremal number O(n 1+ 1 k −ε ). In this paper, we prove both conjectures.

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The rainbow Turán number ex * (n, H) of a graph H is the maximum possible number of edges in a properly edge-coloured n-vertex graph with no rainbow subgraph isomorphic to H. We prove that for any integer k ≥ 2, ex * (n, C 2k ) = O(n 1+1/k ). This is tight and establishes a conjecture of Keevash, Mubayi, Sudakov and Verstraëte. We use the same method to prove several other conjectures in various topics. First, we prove that there exists a constant c such that any properly edge-coloured n-vertex graph with more than cn(log n) 4 edges contains a rainbow cycle. It is known that there exist properly edgecoloured n-vertex graphs with Ω(n log n) edges which do not contain any rainbow cycle. Secondly, we prove that in any proper edge-colouring of K n with o(n r r−1 • k−1 k ) colours, there exist r colour-isomorphic, pairwise vertex-disjoint copies of C 2k . This proves in a strong form a conjecture of Conlon and Tyomkyn, and a strenghtened version proposed by Xu, Zhang, Jing and Ge. Finally, we answer a question of Jiang and Newman by showing that there exists a constant c = c(r) such that any n-vertex graph with more than cn 2−1/r (log n) 7/r edges contains the r-blowup of an even cycle.

Given a graph H, the extremal number ex(n, H) is the largest number of edges in an H-free graph on n vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing K ′ s,t for the subdivision of the bipartite graph K s,t , we show that ex(n, K ′ s,t ) = O(n 3/2− 1 2s ). This proves a conjecture of Kang, Kim and Liu and is tight up to the implied constant for t sufficiently large in terms of s. Second, for any integers s, k ≥ 1, we show that ex(n, L) = Θ(n 1+ s sk+1 ) for a particular graph L depending on s and k, answering another question of Kang, Kim and Liu. This result touches upon an old conjecture of Erdős and Simonovits, which asserts that every rational number r ∈ (1, 2) is realisable in the sense that ex(n, H) = Θ(n r ) for some appropriate graph H, giving infinitely many new realisable exponents and implying that 1 + 1/k is a limit point of realisable exponents for all k ≥ 1. Writing H k for the k-subdivision of a graph H, this result also implies that for any bipartite graph H and any k, there exists δ > 0 such that ex(n, H k−1 ) = O(n 1+1/k−δ ), partially resolving a question of Conlon and Lee. Third, extending a recent result of Conlon and Lee, we show that any bipartite graph H with maximum degree r on one side which does not contain C 4 as a subgraph satisfies ex(n, H) = o(n 2−1/r ).