This should work:

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

[![mesh][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.

If you need a finer mesh you can use:


    ToElementMesh[reg, {{-0.04`, 0.2`}, {0.`, 0.2`}}, 
      "MaxBoundaryCellMeasure" -> 0.005, "MaxCellMeasure" -> 0.00001, 
      "BoundaryMeshGenerator" -> {"RegionPlot", 
        "SamplePoints" -> 41}]["Wireframe"]

[![enter image description here][2]][2]


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