Tur\'an numbers of theta graphs

The theta graph ${\mathrm{\Theta }}_{\mathrm{\ell },t}$ consists of two vertices joined by $t$ vertex-disjoint paths of length $\mathrm{\ell }$ each. For fixed odd $\mathrm{\ell }$ and large $t$, we show that the largest graph not containing ${\mathrm{\Theta }}_{\mathrm{\ell },t}$ has at most $c_{\mathrm{\ell }}{t}^{1-1\mathrm{/}\mathrm{\ell }}{n}^{1+1\mathrm{/}\mathrm{\ell }}$ edges and that this is tight apart from the value of $c_{\mathrm{\ell }}$.