Osaka (KIX) lounge access?
 

There is probably an English page for this, but here: ANA Lounge Information at Kansai International Airport (Kansai International Airport)|International Lounge|ANA it says

Kansai

Who is eligible?
1. Passengers traveling in Business Class on ANA Group-operated flights or flights operated by other Star Alliance member airlines

TK is *A so that means you.

That is the ANA page. There is also this:
Airline Lounges | Kansai International Airport (kansai-airport.or.jp)

btw, Star Silver status won't get you in the lounge but Star Gold status would. In the image at the top of the ANA page the lounge sign has a gold background and a Star Alliance logo which tells you that it is a "Star Gold Lounge" which is what you are looking for with your business class boarding passes (or gold status)

Thanks, mrwunrfl. I was puzzled because on the Turkish Airlines site, it says it has a lounge contract at Haneda, so my interpretation was that there isn’t access anywhere else. But going to the ANA site, it appears to say that people traveling on any Star Alliance flight would have access. I think that’s right. I’m relieved now!

progol, a helpful tool when traveling within the Star Alliance is the "lounge finder" on the Star Alliance website. Put in all your details in the "filters" section (originating airport, destination, FF program and elite level, class of service), and it will tell you which lounges you are eligible to use. In your case, it looks like the ANA lounge. It's been over a decade since we've been to that lounge, so no idea what it's like now.

https://www.staralliance.com/en/lounge-finder