Welcome to the weird and wonderful world of Oracle.
Viewing large amounts of data (especially "wide" data) through sqlplus has always been less than pretty. Even back in the 1990s Oracle rival Ingres had a rather nice isql which made a much better fist of this, although the flipside of that was using isql to spool to a data file (no headers and trimmings, etc) was slightly harder. I think the rather primitive nature of SQLPLus is why TOAD/SQL*Developer etc have become popular.
To make the output easier to read, you need to learn the basics of sqlplus formatting, in particular SET LINES, PAGES, TRIMSPOOL, TAB, and the COLUMN formatting command.
Use COLUMN to control the formatting of each column.
One possible option is to use SET MARKUP and spool to a file, which formats the output as HTML table, but then you need a HTML viewer/browser to view the results.
On PuTTY your options are limited, but if you have xterm and can invoke the browser on Linux, you might find something like a shell script:
#!/bin/bash
sqlplus un/pw @the_file
firefox the_output.html
Contents of the_file.sql:
SET MARKUP ON
spool the_output.html
SELECT * FROM user_objects;
spool off
quit
If you have a share between the Linux system where the the_output.html resides and can mount that on WIndows, you could run the query on Linux with MARKUP oN, spool to the share, then click refresh on the Browser.
Clunky, and not really what you want, but try it and see what you get.