I don't believe you can force screen to overwrite the log. It logs to screenlog.%n by default, where %n is the screen window number (so each window has it's own log). If that file exists, it appends to it.
However, you can tell screen to use a different filename, including a timestamp, so you'll get a new log file each time, but you'll then need to manage the old logs.
In .screenrc you can put the following line,
logfile /path/to/log/screenlog-%n-%Y%m%d-%c:%s
to create log files that include the window number (%n) and the year, month, date, and time.
Alternatively, you could create a bash alias that deletes the log file before running screen, for example,
alias screen='rm /path/to/log; screen'
If you want to affect screen log files in the current directory, just remove /path/to/log/ from the commands above.
Lastly, depending on what you're trying to achieve, the Linux tool script might be more useful than just logging in screen. man script for more information.