This is a case of "show don't tell", or for those that frown upon that phrase, I mean write about the effects of the fear.
Fear reactions tend to be summarized in "flight or fight", which is a nice rhyme, but there is more to it.
Vocally there is screaming, yelling, calls for help, anger and threats: Even threats to the person they fear for: "Don't you dare die! Don't you dare! I swear to God...")
This is part of an overall behavior to try and eliminate the threat. Calls for help, or anger at a messenger: "What kind of fucking doctor are you?! Get me a real doctor god dammit!" Shouting, "Are there real doctors here?!"
Denial: "No no no no this isn't happening you can't do this..."
"This can't be happening, this can't be true, what are we going to do?"
Begging: "Oh please, oh please don't die, don't die!"
Insistence for help and escape: Shouting at a friend "What are we going to do? WHAT?"
Physical agitation; literally running or walking away, violently shaking their head no, literally shaking, an inability to think rationally.
Bargaining: "Please, I'll do anything, anything, just don't let this happen!"
(Added by edit:) I forgot Terror-projections and ramifications: "If Bill... If Bill dies ... What happens to Kate? What do we do?"
Look at this link to the Seven Stages of Grief, and the first 3 of those will be
- Shock and Denial
- Pain and Guilt
- Anger and Bargaining
This is quite close to the various Fear reactions. Now obviously don't include a laundry list of all of them, pick a few instances! In fear, rationality tends to shut down and people act instinctively, including by pushing, hitting and striking out (physically or verbally), or running, or begging, etc.
You don't have to experience all this for yourself, you can often see many good portrayals of fearful people in television or film. You don't want to plagiarize them, obviously but watch what you consider convincing portrayals and note what the fearful characters do and try to characterize the types of things the writers have them say.
You can put those kinds of reactions into your own words, for your character, at the level of fear they are experiencing.