This should work:
ToElementMesh[reg,{{-0.04`, 0.2`}, {0.`, 0.2`}},
"MaxBoundaryCellMeasure" -> 0.005,
"BoundaryMeshGenerator" -> {"RegionPlot",
"SamplePoints" -> 41}]["Wireframe"]
The problem is that "Coninuation" fails (which needs to be investigated) and the "RegionPlot" one does need the "SamplePoints" (think "PlotPloints"). Hope this helps.