I've been climbing (usually once a week) for a couple of years and started lead indoors a about a year ago. I am really scared of falling so much that one week, can maybe do a 6a (6b if am feeling gutsy) lead but then the next time I go climbing, can't even start it and feel like jelly. Sometimes feel like giving climbing up completely - although I keep going back for more. PS. Why can it feel so demoralising?
This is a very common problem. Nobody likes to fall, it goes against all our self preservation instincts but the key for climbing is to somehow override those feelings. Easier said than done!
Some people seem naturally able to push these feelings to the back of their mind and just concentrate on climbing in a relaxed manner but if you are not one of those people, there are many different things you can try.
First, it's worth trying to identify what is the root cause of your fear.
Do you trust your belayer, have you had a jarring fall from someone that has affected your ability to push yourself? It's important to have 100% trust in your belayer, if you don't this will definitely hold you back. So have a chat with your partner, do they know how to give a soft catch/dynamic belay? If not, there is plenty of info about this on the net. If you know your partner can belay well, this can help your confidence a lot.
If you don't feel trust in your belayer is the issue, then the problem lies inside your head.
Are you worried about hurting yourself? As I said this is a perfectly natural fear but in actual reality, very unlikely to happen. I take falls all the time because I like pushing myself and it has only been in the most extreme situations that I have hurt myself.
Saying that, part of being a better climbing is having an awareness of what could happen if you do fall. So it is important when you are climbing, in the back of your mind you are thinking about your body position and how to fall correctly, yes, there is an art to falling! The trick is that when you come into contact with the rock/wall you make yourself as square on as possible, so your knees and legs can absorb the fall, rather than other body parts which don't brace as well. You don't want this to be at the forefront of your climbing experience but it's a good idea to practice when you are climbing easier routes (so you have time to play with it). Ask yourself; what would happen to my body if I fell off now and how can I quickly correct my body position? So that If you fall you do so with as little impact as possible; as I said before, your belayer has a very important role in this process too. You always need to think about having the rope in the correct place, so this can't flip you over. The amount of times I see nasty falls which could have been prevented if the climber had managed to keep the rope in the correct position, rather than twisted round a leg/ankle, which is a recipe for disaster.
Also, wearing a helmet (if you don't already) can certainly help with a feeling of security, even in the wall, it's not that common but if it makes you feel better, why not?
Finally, don't beat yourself up. It may be that you have unrealistic expectations of where you should be with your climbing. Some people annoyingly seem to zoom up through the grades and have a rapid progression but we aren't all like that. So allowing yourself to progress at your natural pace is very important and try not to compare yourself to others.
It's also worth recognising that when you go climbing, what is your mood like? If it feels like you may not want to push yourself that day, then don't. Not every day can be your best day, I've had to learn that one! But sometimes something great happens; the days that start off feeling like they were going to be mediocre can end up being fantastic. And this is because you have just relaxed into not expecting too much and just naturally ended up being able push harder than you thought, it just creeps up on you.
I hope there is something in here that can help you and you don't give up on climbing. It sounds like you have an affinity with it and it would be a shame to lose that.