The line

shpSavedLayer.setCrs(projectCrs)

overrides your layers' CRSs regardless of their coordinates' actual CRS. Don't do that and your map should look as expected.

You only need to use that method if you want to fix a wrongly detected or missing CRS. You must not use it if your data is valid and correct.

The line

shpSavedLayer.setCrs(projectCrs)

overrides your layers' CRSs regardless of their coordinates' actual CRS. For that to work properly you must know the actual CRS of your data.

The data you are using https://data.cityofnewyork.us/Housing-Development/Public-Use-Microdata-Areas-PUMA-/cwiz-gcty comes with a PRJ file of

GEOGCS[
  "WGS84(DD)", 
  DATUM["WGS84", SPHEROID["WGS84", 6378137.0, 298.257223563]],
  PRIMEM["Greenwich", 0.0], UNIT["degree", 0.017453292519943295], 
  AXIS["Geodetic longitude", EAST], AXIS["Geodetic latitude", NORTH]
]

While this is actually a definition for WGS84 (EPSG:4326, at least the parameter values are the same), QGIS fails to properly process this as you must have noticed. But most importantly: That data is not in EPSG:3395.

To fix this, you need to set the layer's CRS separately:

layerCrs = QgsCoordinateReferenceSystem.fromEpsgId(4326)
shpSavedLayer.setCrs(layerCrs)

For the project/canvas you can use whatever CRS you like, EPSG:3395 8WGS 84 / World Mercator) would be a distorted view as well, I would recommend EPSG:26918 (NAD83 / UTM zone 18N) for optimal results.