There are multiple mistakes in the shared code, primarily -
- s++;move the pointer till- '\0'. It should be brought back 1 unit to
point to actual string by putting- s--. Other wise the copied one will start with '\0' that will make it empty string.
- Magic numbers 20 and 13. where in malloc()1 + length of s should be
sufficient instead or 20. For 13 just move a unit ahead and put'\0'
However, using string.h library functions() this can be super easy. But I think you are doing it for learning purpose.
Therefore, Corrected code without using string.h lib function() should look like this:
char *reverseStr(char s[])
{
    printf("Initial string is: %s\n", s);
    int cCounter = 0;
    while(*s != '\0')
    {
        cCounter++;
        s++;
    }
    s--; //move pointer back to point actual string's last charecter
    printf("String contains %d symbols\n", cCounter);
    char *result = (char *) malloc(sizeof(char) * ( cCounter + 1 ));
    if( result == NULL ) /*Check for failure. */
    {
        puts( "Can't allocate memory!" );
        exit( 0 );
    }
    char *tempResult = result;
    for (int begin = 0; begin < cCounter; begin++)
    {
        *tempResult = *s;
        s--; tempResult++;
    }
    *tempResult =  '\0';
    //result[cCounter+1] = '\0';
    return result;
}
Calling from main
int main()
{
    char testStr[] = "Hello world!";
    char *pTestStr;
    puts("----------------------------------");
    puts("Input a string:");
    pTestStr = reverseStr(testStr);
    printf("%s\n", pTestStr);
    free(pTestStr);
}
Output
----------------------------------
Input a string:
Initial string is: Hello world!
String contains 12 symbols
!dlrow olleH
As per WhozCraig suggestion just by using pointer arithmetic only - 
char *reverseStr(const char s[])
{
    const char *end = s;
    while (*end)
        ++end;
    char *result = malloc((end - s) + 1), *beg = result;
    if (result == NULL)
    {
        perror("Failed to allocate string buffer");
        exit(EXIT_FAILURE);
    }
    while (end != s)
        *beg++ = *--end;
    *beg = 0;
    return result;
}