Sam,
I just joined this forum to write you this. I was in your shoes just a few months ago. Now I try to fly every chance I get. There's still a healthy sense of fear, but it's more of respect than true fear. It's that sense which makes me double & triple-check things, especially when flying alone. I don't think this is a bad thing. Complacency in aviation is a bad thing. I also fly with a GoPro stuck over my shoulder so I can see the instruments, controls, out the window and do a self de-brief after every flight and take notes.
I myself am currently a student pilot. I started this with zero self-confidence. More like negative self-confidence. Crashing and damaging the plane were both my biggest fears and were getting in the way of flying. When landing I'd instinctively pull back as the ground started to appear rapidly approaching. At 10 hours and ~20 landings, my CFI was pushing me to solo. "H*** NO!" was my response. He had confidence in my abilities even if I didn't.
Over the New Year's holiday I was out of town and found a CFI who owned a tail-dragger. I spent two days with him. In that two days I did ~35 landings, stalls, slow flight, etc. and full-on spin training (which he didn't tell me about). If you think the ground rapidly approacing is bad, it's even worse when your CFI decides to do unannounced spin training over the Atlantic Ocean and all you can see out the front window is water. We kept doing that until it became instinctive for me to recover. Oh, I also set my personal minimum for bottle-to-throttle time of 24 hours. 10 hours may have been legal and good enough for straight and level flying, but New Year's eve + unexpected spin training taught me that 10 hours wasn't enough in case of upset.
For me, the key to the fear of landing was knowing where to look and flying by the numbers.
First, where are you looking? Your gaze needs to change as the phases of the landing change. Let's assume you're doing a straight-in approach. At 10 miles you're still looking waaay out there until you've positively identified the field. Then you should be looking at the PAPIs/VASIs. Generally they're pretty close to the aiming point markers. As I'm getting under 100' AGL, I'll take a *quick* glance out to the side window for perspective. I'll usually sneak another one around 50' AGL. As you're about to reach the fence you need to be looking about the middle of the runway. Once you cross the fence you need to start your flare and gradually look (and stay focused) on the very far end of the runway. Don't look any closer than the VERY END of the runway until your wheels are firmly on the ground. *FLY* the plane in this, don't think about it as landing. Remember, even once you get your license, go-around should be your default. Landing is optional. Don't like it? Firewall the throttle (gently) and go around. There's no shame in a go-around. Get the airplane back up to pattern altitude, regroup, and try it again. If things are looking decent (not necessarily great), continue getting closer to the ground.
Second, flying the numbers is key. I had the nasty habit of coming in with "a little extra" altitude and speed, not wanting to stall the aircraft. This is self-defeating and makes the situation worse. The extra speed/altitude just increases your chances of a bounced landing or a very floaty one. Unless the big fan up front stops spinning more speed/altitude is always available in the throttle which your right hand should be glued to during final approach. When you have the speeds spot-on, landings become a non-event. I'm a new user so it won't let me post links, but Google Hekster Traffic Pattern. Take that diagram to your CFI and have them mark it up to match the speeds/RPMs/configuration/altitude FPM for each phase for your aircraft.
I finally felt like I was ready to solo with about 50 landings under my belt. Unfortunately the weather was uncooperative for almost a month (crosswinds too high and/or storms), so I had more dual time and had to wait to finally solo. We used the time to practice engine-outs, navigation, and all sorts of emergencies.
Finally, I soloed. Unlike what you see on YouTube, it was completely anti-climactic. On the crosswind leg, the left vacuum pump light started flickering. By the time I turned to the downwind leg, it died. The right pump decided to join it by mid-field leaving me with no vacuum system. I wasn't nervous; I was angry. I waited over a month after I felt ready now the aircraft was going to try to stop it? I pondered the implications. Vacuum powers the directional gyro and attitude indicator (artifical horizon) on a C172. I was flying VFR, in the pattern, on a clear day, at my home airport. Turn coordinator is electric and would provide bank angle info, the whiskey compass still worked, and I had a pair of working eyes. Vacuum? I didn't need any stinkin' vacuum for this flight. I did my 3 touch & goes and landed, taxiied back to the hangar, cursed at the plane as I wrote up the squawks and that was it. Just another day in the office, flying the pattern, landing, taking off, etc. The vacuum pumps dying didn't even remotely phase me.
Looking back, my CFI was right... I did have the technical skills at 10 hours. I didn't have the confidence, but I was capable of putting the plane down safely on the runway at that time even if I didn't feel comfortable with it. Trust your CFI. They're not going to let you hurt yourself.
If you're looking for greasers, just remember all of the hard/bad landings you've experienced in airliners. Those captains have 1,500+ hours, even 10,000+ hours, all sorts of technology in the cockpit, easily hundreds of thousands of dollars in training including full motions sims, and they still borked the landing that badly...and he didn't bend the plane and everyone walked away safely. For what it's worth, I know more than one airline pilot who felt like you in the beginning of their training.
Don't give up. Don't be afraid to try a different CFI if it's not clicking.
My suggestions:
1) A plane it not a car. Don't expect it to behave like one nor expect your driving skills to transfer over. My CFI said he's found youth do better in training because they don't have decades of driving experience. Wolfgang Langewiesche of Stick and Rudder fame compares the airplane more like a horse.
2) Don't worry about soloing right now. For that matter, don't worry about crashing. You have a fully qualified pilot sitting next to you. He cares about his life too. He won't let you do anything that would put his life in jeopardy.
3) Get that diagram I mentioned earlier and mark it up. Have the CFI fly the pattern with you calling out the info and watching him fly it. Do a touch-and-go. On the next one, you operate the throttle / flaps and nothing more, let him control the ailerons & elevator, again, going by the diagram and his instructions. Build your sucessful landings step-wise. When you're getting close to operating all of the controls, see if you can extend your downwind and do a long final approach. Use the time to absolutely nail a stable approach. Stable approach = checklist complete, on glide-slope, correct speed, correct power, correct configuration. and remember, you're planning on going around. If it feels even remotely decent keep getting closer to the ground. You don't have to commit to a landing until you've crossed your normal take-off abort point with your wheels still on the ground.
4) Dial up the ILS. Yes, you're a VFR pilot and yes the PAPIs should give you this information, BUT it's another tool in your arsenal. Especially if you're flying a long final approach, this will give you additional confirmation and confidence that you're on the right path horizontally and vertically. With those locked in, all that's left is airspeed to worry about.
5) Get a GoPro. Record EVERY flight. After the flight, de-brief even if just by yourself. Do watch it on a larger screen so you can see the instruments and flight controls. You'll be surprised at some of the mistakes you made, but you'll also see all that you're getting right. There's even a few small things I'm better at than my CFI and he's been teaching for 20 years.
6) Fly often. How many times a week are you flying? Shoot for at least twice a week, at a minimum. Remember what the first day of each school semester felt like? The awkwardness? The confusion? How did you get over that? Subconsciously by just doing it daily. After the first month of the semester, I bet everything felt pretty normal.
7) Go ahead and get renter's insurance. If anything happens to the plane, it's the insurance co's problem. My CFI actively teaches this. Engine out? Insurance co owns the aircraft now, screw the plane, just use it as a tool to get you down safely. Bend a prop? That's the insurance co's problem, not yours.
8) How long are you flying? I'd push for 1.5-2 flight hours per session.
9) Don't forget how far you've come. Look at all of the things you found overwhelming at first. Pre-flighting, trying to get that ornery engine to start, runway markings, radio communications, taxiing. This too shall pass, just like all of those obstacles before you.
10) With aviation, you never stop learning.