diff --git a/CHANGES b/CHANGES
index 5c1650c28..f0d4e8db5 100644
--- a/CHANGES
+++ b/CHANGES
@@ -38,6 +38,9 @@ Bugs fixed
 * #3873: Failure of deprecation warning mechanism of
   ``sphinx.util.compat.Directive``
 * #3874: Bogus warnings for "citation not referenced" for cross-file citations
+* #3860: Don't download images when builders not supported images
+* #3860: Remote image URIs without filename break builders not supported remote
+  images
 * #3833: command line messages are translated unintentionally with ``language``
   setting.
 * #3840: make checking ``epub_uid`` strict
diff --git a/sphinx/builders/__init__.py b/sphinx/builders/__init__.py
index fdf524bf6..007964e82 100644
--- a/sphinx/builders/__init__.py
+++ b/sphinx/builders/__init__.py
@@ -22,6 +22,7 @@ from six import itervalues
 from docutils import nodes
 
 from sphinx.deprecation import RemovedInSphinx20Warning
+from sphinx.environment.adapters.asset import ImageAdapter
 from sphinx.util import i18n, path_stabilize, logging, status_iterator
 from sphinx.util.osutil import SEP, relative_uri
 from sphinx.util.i18n import find_catalog
@@ -200,6 +201,7 @@ class Builder(object):
     def post_process_images(self, doctree):
         # type: (nodes.Node) -> None
         """Pick the best candidate for all image URIs."""
+        images = ImageAdapter(self.env)
         for node in doctree.traverse(nodes.image):
             if '?' in node['candidates']:
                 # don't rewrite nonlocal image URIs
@@ -210,7 +212,8 @@ class Builder(object):
                     if candidate:
                         break
                 else:
-                    logger.warning('no matching candidate for image URI %r', node['uri'],
+                    logger.warning('no matching candidate for image URI %r',
+                                   images.get_original_image_uri(node['uri']),
                                    location=node)
                     continue
                 node['uri'] = candidate
diff --git a/sphinx/transforms/post_transforms/images.py b/sphinx/transforms/post_transforms/images.py
index e1cc9f78b..c57355774 100644
--- a/sphinx/transforms/post_transforms/images.py
+++ b/sphinx/transforms/post_transforms/images.py
@@ -20,7 +20,7 @@ from sphinx.transforms import SphinxTransform
 from sphinx.util import logging, requests
 from sphinx.util import epoch_to_rfc1123, rfc1123_to_epoch
 from sphinx.util.images import guess_mimetype, get_image_extension, parse_data_uri
-from sphinx.util.osutil import ensuredir
+from sphinx.util.osutil import ensuredir, movefile
 
 if False:
     # For type annotation
@@ -57,7 +57,9 @@ class ImageDownloader(BaseImageConverter):
 
     def match(self, node):
         # type: (nodes.Node) -> bool
-        if self.app.builder.supported_remote_images:
+        if self.app.builder.supported_image_types == []:
+            return False
+        elif self.app.builder.supported_remote_images:
             return False
         else:
             return '://' in node['uri']
@@ -67,6 +69,8 @@ class ImageDownloader(BaseImageConverter):
         basename = os.path.basename(node['uri'])
         if '?' in basename:
             basename = basename.split('?')[0]
+        if basename == '':
+            basename = sha1(node['uri']).hexdigest()
         dirname = node['uri'].replace('://', '/').translate({ord("?"): u"/",
                                                              ord("&"): u"/"})
         ensuredir(os.path.join(self.imagedir, dirname))
@@ -94,6 +98,14 @@ class ImageDownloader(BaseImageConverter):
                     os.utime(path, (timestamp, timestamp))
 
                 mimetype = guess_mimetype(path, default='*')
+                if mimetype != '*' and os.path.splitext(basename)[1] == '':
+                    # append a suffix if URI does not contain suffix
+                    ext = get_image_extension(mimetype)
+                    newpath = os.path.join(self.imagedir, dirname, basename + ext)
+                    movefile(path, newpath)
+                    self.app.env.original_image_uri.pop(path)
+                    self.app.env.original_image_uri[newpath] = node['uri']
+                    path = newpath
                 node['candidates'].pop('?')
                 node['candidates'][mimetype] = path
                 node['uri'] = path
@@ -108,7 +120,9 @@ class DataURIExtractor(BaseImageConverter):
 
     def match(self, node):
         # type: (nodes.Node) -> bool
-        if self.app.builder.supported_data_uri_images:
+        if self.app.builder.supported_remote_images == []:
+            return False
+        elif self.app.builder.supported_data_uri_images is True:
             return False
         else:
             return 'data:' in node['uri']
diff --git a/sphinx/util/images.py b/sphinx/util/images.py
index 2d228e66d..eba295a3c 100644
--- a/sphinx/util/images.py
+++ b/sphinx/util/images.py
@@ -120,3 +120,17 @@ def parse_data_uri(uri):
 
     image_data = base64.b64decode(data)
     return DataURI(mimetype, charset, image_data)
+
+
+def test_svg(h, f):
+    """An additional imghdr library helper; test the header is SVG's or not."""
+    try:
+        if '<svg' in h.decode('utf-8').lower():
+            return 'svg+xml'
+    except UnicodeDecodeError:
+        pass
+
+
+# install test_svg() to imghdr
+# refs: https://docs.python.org/3.6/library/imghdr.html#imghdr.tests
+imghdr.tests.append(test_svg)