@Schaick:  From your reply to MadBill, it sounds like what you might be working on is what I would call "fluency".  The ability we use in spoken language to select words and phrases that fit, and doing it "on the fly" as we do in verbal conversation.

I am pretty much always working on that for violin.  I can do it okay on guitar and some other instruments, but the violin is still kinda like... Well, there is a saying some of the older folks around here used to say, who were of Swedish descent..

"If I could only say in English what I am thinking in Swedish.."

But what I have found to be of the most use for that is doing a fair amount of playing around against a backing track with a simple repetitive pattern to it.  I use this one, sometimes..

If you are maybe a bit new to improvising, at first it can be easier to stick to a simple "pentatonic" scale.  For the key of Am, as this example, it would be just the notes A C D E G.  "Penta Tonic" because it has 5 tones.

Why those 5, in particular?  Well, they are the ones that are pretty 'safe' to play against a standard "vanilla" chord progression like this one.  Any of those notes will sound at least "not bad" against almost any moment of that backing track (or any other piece written in anything like a standard sort of chord progression for A minor or C major).  So it makes for some safe "baby steps" to just play around in and see what you can come up with.

Take it slow and easy at first and remind yourself that silence, like a pause or space, is as much a part of music as notes are.  No harm at all in listening for a few beats while you decide what note to play next when improvising.

If you can play through the whole track for the entire 20 min and sound great and brilliant the entire time, please hit yourself over the head, tie yourself up and lock yourself in a closet, before you make the rest of us look bad.  More likely, things you try will sound kind of boring or not particularly good (though not "wrong) part of the time, but part of the time you'll hear yourself doing some combinations and changes that actually sound pretty darn good.  When you find those, repeat them several more times, so your fingers can learn the moves and you burn it into memory a bit.

20 min is actually quite a long time to have to improvise.  In real life musical situations, it almost never happens.  But you can build up a vocabulary of moves/sounds this way so that when you are in a more realistic situation where you may have to "take the lead" for a minute or less, you'll already have "vocabulary" and "phrases" you've found that you can use.   Also, since the chord pattern repeats a lot of times over that 20 min, you get to practice it "coming around to you" a lot, which may help with that anxiety a bit.

You may be thinking "But we don't do any songs that are exactly like that backing track".  Probably not, but the chord changes are very typical, and what notes sound good against the chords and changes are something you can use in a lot of different songs.

The songs you are currently doing might not be strictly out of the "pentatonic scale" that I gave as an example.  But if you're doing Celtic, folk, traditional, pop, country, rock, blues, jazz or etc, a surprising amount of the songs you do will contain at least some passages that are that pentatonic scale.

For example, you can play the entire verse part of "Oh Susanna" using only those 5 notes.  (C D E G G A G E C D E E D C D)  Lots of other songs or parts of songs as well.

As you get comfortable with improvising, you can add in other notes like the B and F to get a full minor scale and etc. 

But to get back to the point, if I understand what you're trying to do correctly, what I have found to be helpful for it is to include improvising into your practice routine.  Doing it for a longer stretch than you ever have to "take the lead" for makes it easier when you take the lead in realistic situations.  Your fingers already will have learned plenty of things to do that can sound good, and will have already practised a lot of the note changes you are likely to need.  You can get used to your time to play "coming around" so you can get to feel easy about it.  You also will discover some tasty bits you can use as fill or embellishments that are "purely you" that can fit well with the melodies of the actual songs you may be playing.