This should work:

    ToElementMesh[reg,{{-0.04`, 0.2`}, {0.`, 0.2`}},
    "MaxBoundaryCellMeasure" -> 0.005, 
      "BoundaryMeshGenerator" -> {"RegionPlot", 
        "SamplePoints" -> 41}]["Wireframe"]

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


The problem is that "Coninuation" fails (which needs to be investigated) and the "RegionPlot" one does need the "SamplePoints" (think "PlotPloints"). Hope this helps.

  [1]: https://i.sstatic.net/UObX3.png