There should be a nice proof, but here is a reference that proves something stronger and weaker.  This [paper](http://oai.cwi.nl/oai/asset/1443/1443A.pdf) by de Graaf and Schrijver proves that every graph embedded on the torus with face-width at least $n \geq 5$, contains the toroidal $\lfloor 2n/3 \rfloor$-grid as a minor.  Note that the toroidal $\lfloor 2n/3 \rfloor$-grid has (almost) $8n^2/9$ edges.  So any graph on the torus with face-width at least $n$ has at least (almost) $8n^2/9$ edges, which is pretty close to the conjectured answer of $n^2$.