Why am I getting "line 1: $' :\r': command not found"?

This is due to the script having been saved by an editor that uses DOS line-endings (such as Notepad++, for example). You will have to remove these from your scripts.

To do this, either use dos2unix on the script file, or

$ tr -d '\r' <script >script.new && mv script.new script

(this will remove all carriage returns from anywhere in the script)

As for the code in the script:

:
iter=1
if [ -f iter.txt ]
   then rm ./iter.txt 
fi  

This should possibly look something like

iter=1
if [ -f "./$iter.txt" ]; then
   rm "./$iter.txt"
fi

This removes the file 1.txt from the current directory, if it exists. The : command does nothing (almost) and may be removed. The iter variable's value should be used as $iter, and be quoted. And then I'm possibly being more explicit than needs be by using ./ to say that the file needs to be found in the current directory.

If you're planning to turn this into a loop (here, deleting all files 1.txt, 2.txt, ..., 10.txt):

iter=1
while [ "$iter" -le 10 ]; then
    if [ -f "./$iter.txt" ]; then
        rm "./$iter.txt"
    fi
    iter=$(( iter + 1 ))
done

Or, if we feel sneaky/lazy,

rm -f {1..10}.txt

in shells that understands brace expansion.

Your scripts have the wrong line-endings. Windows uses CR+LF (\r\n), Unix uses LF (\n), and Classic MacOS uses CR (\r). You'll need to change the line-endings on all affected files, either with a text editor or a standalone tool like dos2unix.

FTP and version control systems like SVN tend to be able to convert the line endings appropriately, so you might want to look into those options for transferring files in the future.

If these files aren't being transferred, but just used on a single machine, then you'll want to find the settings for line endings in your preferred text editor. Most that are advertised as "for programmers" will have such an option.

As a supplemental answer let me add a couple notes/tips...

First, for regular text files you can easily determine whether they have Unix or DOS line ending using the file command. For example:

$ file test.txt
test.txt: ASCII text, with CRLF line terminators
$ dos2unix test.txt
dos2unix: converting file test.txt to Unix format...
$ file test.txt
test.txt: ASCII text

Note my use of dos2unix. I highly recommend you install it. Depending on how you use Cygwin you may find you need to do this conversion often enough that the convenience is welcome. More importantly there is no chance of error like you get with the manual edits discussed here.

To install start the main Cygwin executable. That's the one with the name like setup-x86_64.exe or something along those lines. Make sure you see a screen like this:

Since you have already done the primary install you should be able to click Next all the way until you get the selection screen. Select Full in the dropdown and enter "dos2" in the search box and choose the tool as shown here:

Then click Next again and let the installation complete. That's it. Have fun. Cygwin, with the exception of this occasional inconvenience, is an awesome package.

If not dos2unix, there is also 'flip'

$ flip -u yourfilename.txt

will flip it to using unix line endings.

I use Notepad++ to edit my bash (.sh) scripts by connecting to the server by WinSCP
(To set up WinSCP editor to Notepad++ : https://winscp.net/eng/docs/ui_editor_preferences )

This is very practical for opening files from servers to edit/save instead of "Vim" editor.
When you open the file in Notepad++, double-click on the 7th column at the status bar at the bottom and select "Unix(LF)".

Also, you have to change the 8th one by Encoding -> Encode in UTF-8 in the top bar.

You don't need to change the actual file's line encoding to have Cygwin's Bash happily work with Mac and Windows end-of-line encodings. There's an option you just need to flip on.

Just update your ~/.bash_profile so it contains

export SHELLOPTS
set -o igncr

See https://ptolemy.berkeley.edu/projects/chess/softdevel/faq/5.html