Bash Script – Find -exec with Variable Expansion

I'm trying to run a command similar to the one below in a bash script. It should search through all subfolders of $sourcedir and copy all files of a certain type to the root level of $targetdir.

#!/bin/bash

# These are set as arguments to the script, not hard-coded
sourcedir="/path/to/sourcedir"
targetdir="/path/to/targetdir"

find "$sourcedir" -type f -name "*.type" -exec sh -c 'cp "$1" "$2/`basename "$1"`"' "{}" "$targetdir" \;

This seems to be pretty close, except that {} isn't being passed in as $2 to -exec sh -c ...

I would like to do this as close to the "right way" as possible, with tolerance for special characters in filenames (specifically single quote characters).

Edit: I see people suggesting using xargs or argument chaining. I was under the impression that this is only ok for a limited number of arguments. If I have, for example, thousands of .jpg files I'm trying to copy from a number of gallery directories into a giant slideshow directory, will the solutions chaining arguments still work?

Edit 2: My problem was I was missing a _ before my first option to sh in the -exec command. For anyone who is curious about how to make the find command work, add the _ and everything will be fine:

find "$sourcedir" -type f -name "*.type" -exec sh -c 'cp "$1" "$2"' _ "{}" "$targetdir" \;

I have accepted an answer below though because it accomplishes the same task, but is more efficient and elegant.