Yes, you are definetly normal.
The reason why "people do presentations like its a walk in the park" is because they have high self-esteem. Usually they have a big mouth to begin with, and they rarely close it. So, doing a presentation for them is no different than what they do on a daily basis. Don't worry, like knotme said, public speaking is just a skill. It means that if you are willing to put the effort, you can be as good as those people who have this skill naturally.
Before, I used to write and memorize all my presentations, then wonder why teachers gave me only passing marks

I am not the kind of person who can go in a group and knows everyone within 10 minutes. In class, I usually don't speak unless I am asked a question. Still, I can do a presentation better than the loudest speaker in the class. Why? Because I mastered the skill.
I will give you some pointers that I consider a must. You must know your subject upside down. I literally mean it. If someone ask you a question on your subject, you should be able to answer right away. Obviously, you must also be a good researcher, so it's an other skill to learn. I stress on this "know your subject" because it will give you a boost (an unimaginable boost) in your confidence. Once you know your subject, you do an outline of your presentation. You don't go into details. You don't want to see long paragraphs of what you are going to say

Just a list of the points you are going to address, to know how the presentation is going to flow. Then, during the presentation, your are going to improvise on those points. But it's GOING TO BE EASY because you are an expert in your subject, and you can talk about it for days.
Another thing to boost your confidence is to dress to impress. If you look at public speakers, they always dress in a suit--wonder why? Well, you should do so too. You can't overdress, but the opposit is possible. Good cloths projects a good image of yourself to others.
Finally, eye contact is very important. Never stare at a person, that is, never look at a person more than 3 seconds in a row. The way I do eye contact is like so. Picture that the square below is the audience infront of you.
2###3
#####
##5##
#####
1###2
The numbers represent where you should look at and in what order. I don't have a definit rule on how long you should look at each number. Keep in mind that the full cycle shouldn't be too quick nor too slow. If you do it in front of a mirror, you will notice that if you move your head too fast, it doesn't seem natural. If you move it too slow, you look like staring (bad!).
I invite you to go and research more on public speaking. What I talked about is just the peak of the iceberg.
On a last note, the signs of nervousness you mentioned, I still have them when I do a presentation. However, they disappear after 30 seconds or so.