You would need to exec
the script itself, like so
#!/bin/bash
echo "Doing my stuff here!"
# Show message
read -sp "Press ENTER to re-start"
# Clear screen
reset
# Re-execute the script
exec bash ./run_this.sh
exec
does not work with shell scripts, so you need to use execute bash
instead with your script as an argument.
That said, an in-script loop is a better way to go.
while :; do
echo "Doing my stuff here!"
# Show message
read -sp "Press ENTER to re-start"
# Clear screen
reset
done