The cow mesh is an example of a "broken" mesh. Try 
    
    mesh =  RepairMesh[mesh]

before sending it to `ToElementMesh`. 


Among other nice meshes, you can find a free and "clean" cow mesh also on Keenan Crane's homepage:

https://www.cs.cmu.edu/~kmcrane/Projects/ModelRepository/

This is the model (without texture):

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/qgWXU.png