Ensure your libcurl.so module is in the system library path, which is distinct and separate from the python library path.
A "quick fix" is to add this path to a LD_LIBRARY_PATH variable. However, setting that system wide (or even account wide) is a BAD IDEA, as it is possible to set it in such a way that some programs will find a library it shouldn't, or even worse, open up security holes.
If your "locally installed libraries" are installed in, for example, /usr/local/lib, add this directory to /etc/ld.so.conf (it's a text file) and run "ldconfig"
The command will run a caching utility, but will also create all the necessary "symbolic links" required for the loader system to function. It is surprising that the "make install" for libcurl did not do this already, but it's possible it could not if /usr/local/lib is not in /etc/ld.so.conf already.
PS: it's possible that your /etc/ld.so.conf contains nothing but "include ld.so.conf.d/*.conf". You can still add a directory path after it, or just create a new file inside the directory it's being included from. Dont forget to run "ldconfig" after it.
Be careful. Getting this wrong can screw up your system.
Additionally: make sure your python module is compiled against THAT version of libcurl. If you just copied some files over from another system, this wont always work. If in doubt, compile your modules on the system you intend to run them on.