I cant easily come up with a verbal expression I can use when I decided to stop talking about something because it's a sensitive topic, for example. One expression I can think of is 'I'll stop talking about it here', but it feels somewhat redundant and too much decisive. Another expression is 'I'll save my words', but is it commonly used?
There are many options available to you here, and they often differ in the reasoning for not saying something.
I'll bite my tongue
This is a colloquialism used when you might have otherwise said something negative or incriminating, normally involving somebody else.
It's not my place to say
If the thing in question is about somebody else and it's of a personal nature, you might choose this.
I'd rather not talk about it
You might use this option if the thing involves you or upsets/embarrasses you.
Now is not the time to talk about this
This one's pretty self-explanatory, and can also imply that this is neither the time nor place to talk about the thing.
I don't have [much/anything else] to say about it
A straight-forward approach that should help end a discussion.
(Anyway,) let's move on
To move the conversation forward, and hopefully avoid any discussion at all about it.
Again, there are many options available to you depending on context.
To add on to Dan's excellent answer: I agree that there are many options, and the nuances of each vary depending on what you want to say. Here are some others:
I'll just leave it at that
I think I've said enough on that topic (for now).
Well, you've heard (enough about) my opinion on the subject.
I've blabbered/babbled/prattled/gabbed on long enough.
I'll just shut up now.
Enough said. (or 'nuff said)
The following bunch of examples are probably the simplest phrases you can use to let your interlocutor know that you realize you're breaching his or her discussional comfort zone and thus will not continue discussing the issue that makes them feel uncomfortable any further:
I see that talking about this makes you feel uncomfortable. Let's just switch the topic.
You don't feel comfortable talking about this? Oh, I see. Let's just change the topic.
The death of your father is a sensitive topic to you? Then, let's not touch on that.
You don't feel comfortable talking about this? Then, let's not touch that subject.
I understand that you don't feel comfortable talking about this. Let's move on to something else, shall we?