Do you think this is normal?
Yes. I'm nearly halfway through my PPL course. I'm wondering about much the same things. You can practice engine failures but doing it for real will be a totally different experience. I'm not at all sure I'd make a good emergency landing, but I will just have to try my best. Same thing about landing at an unfamiliar airport.
Also confidence goes up and down again. Today I practised crosswind landings in a tailwheel aircraft. Last week I made some decent landings and thought I understood how to land a taildragger in a crosswind. Today I did exactly the same but somehow found it much harder. I just need to practice untill I don't worry about it anymore and then I'm ready to add another thing to the workload.
How often do you fly? I found it makes a lot of difference if I fly two times a day.