Yes, of course words hurt
"Sticks and stones may break my bones but words will never hurt me" is a common meme, buy everyone knows it is false. Of course words hurt. Even the people who repeat that meme know it is false.
Many people have been deeply hurt by harsh words from a loved one. Many people have been hurt by being berated at work or school. When someone feels unappreciated because of constant criticism and no appreciation for good done, it can be deeply deflating. Libel and slander have ruined many good names and damaged or destroyed people professionally. Constant teasing and mocking have driven many teenagers and school children to depression and even suicide.
Those who flippantly deny the power of words deny Scripture itself. James 3 warns us of the danger of sinful speech, and even says the tongue is set on fire of Hell. Scripture warns us of the power of our speech in Psalm 10:7, Proverbs 12:18 and Proverbs 18:21. Scripture teaches in Proverbs 21:23 that controlling our speech can keep us from much grief. Which should you trust: God's Word or a trite saying of men?
This does not mean we should be snowflakes. We can all think of many examples of political correctness, where people are "triggered" or "offended" by opinions they disagree with or even facts that they do not want to hear. There are times when the "offended" person is wrong, and nothing that the speaker said was wrong in content or tone. What we should not do, however, is deny reality and we certainly should oppose the clear teaching of the Bible. The reality is that words can and do cause a great deal of harm.