The reference given in the Wikipedia article on linkless embedding for the $4n-10$ bound on the number of edges in a linkless embeddable graph is Mader, W. (1968), "Homomorphiesätze für Graphen", Mathematische Annalen 178 (2): 154–168, doi:10.1007/BF01350657. Apparently Mader proves this bound more generally for $K_6$-minor-free graphs. As the Wikipedia article also states, the example of apex graphs shows that this is tight.

The reference given in the Wikipedia article on linkless embedding for the $4n-10$ bound on the number of edges in a linkless embeddable graph is Mader, W. (1968), "Homomorphiesätze für Graphen", Mathematische Annalen 178 (2): 154–168, doi:10.1007/BF01350657. Apparently Mader proves this bound more generally for $K_6$-minor-free graphs. As the Wikipedia article also states, the example of apex graphs shows that this is tight.

As for the Colin de Verdière invariant: it is known to be at least the size of the largest clique minor minus one – e.g. see http://homepages.cwi.nl/~lex/files/cdvsurvey_new2.pdf – and combining this with the Kostochka/Thomason results cited by Tony Huynh shows that the edge density can grow at most slightly superlinearly as a function of the CdV invariant.