This is a good question, and one that speaks to customer service so I love it!
I’m assuming you’re using the phrase in instances like, “It turns out that your order isn’t in the warehouse due to an internal error.” Or, “It turns out, we no longer offer the special BBQ burger.”
In these cases, it sounds like you’re the bearer of bad news without taking responsibility because it’s just the way it has “turned out,” rather than accepting your role in the bad news.
To be a little more customer friendly and to soften the blow, as you say, you could try:
“I’m sorry to say that...
“I’m afraid we (mishandled/miscommunicated...) this and can offer you this...”
“Apparently, we no longer...”
“I’ve learned that we ...”
Also, words are only half the equation. Consider your non-verbal communication too, including eye contact, smile, volume in terms of speaking softer or slower, which all convey a warmer, more approachable stance.
I am a writing coach and work with clients on written and spoken signage and memos if that helps.
Feel free to view my profile and let me know if you have any other questions.
Hope that helps.