import re
p = re.compile(r'<a\s+href\s*=\s*"([^"]+).*')
test_str = ["My website is <a href=\"WEBSITE1\" target='_blank'><u>WEBSITE1</u></a>", "The link is <a href=\"LINK1\" target='_blank'><u>LINK1</u></a>"]
for x in test_str:
print(re.sub(p, r"\1", x))
aW1wb3J0IHJlCnAgPSByZS5jb21waWxlKHInPGFccytocmVmXHMqPVxzKiIoW14iXSspLionKQp0ZXN0X3N0ciA9IFsiTXkgd2Vic2l0ZSBpcyA8YSBocmVmPVwiV0VCU0lURTFcIiB0YXJnZXQ9J19ibGFuayc+PHU+V0VCU0lURTE8L3U+PC9hPiIsICJUaGUgbGluayBpcyA8YSBocmVmPVwiTElOSzFcIiB0YXJnZXQ9J19ibGFuayc+PHU+TElOSzE8L3U+PC9hPiJdCgpmb3IgeCBpbiB0ZXN0X3N0cjoKCXByaW50KHJlLnN1YihwLCByIlwxIiwgeCkp