I get shy when I sometimes can't add for example 15 + 23 very quickly. It does not happen just with these numbers but also with others except very simple ones such as 4 + 4 or 10 + 14, these are easy. So what do you recommened me to do exactly?
The more you practice adding, the easier it will become.
Sometimes, when we're shy, or embarrassed about something, we find ways to avoid doing it; that just makes it more stressful when you NEED to add...And I'm assuming you have trouble especially when you need to add "in your head", like trying to add the prices of items you've got in your shopping cart (few of us have paper and pencil ready to go at times like that!)
You might find the Khan Academy helpful: if you scroll to "Developmental Math", you'll see a series of lessons, including lessons, on adding (addition). Each lesson involves a 12 minute video lecture, with exercises for practice. Use of this site is free...no strings attached! It might be helpful to view the "menu" of lessons provided under "Developmental Math": arithmetic (like adding) in decimal notation, which comes in handy when dealing money, usually expressed in dollars (digits to left of decimal point) and cents (figits to the right of the decimal point). And many more lessons...!
More immediately: Sometimes developing "tricks" can help. E.g., it can help to transform more complicated combinations of numbers to numbers that are easier for you to add. For example, consider the example you provided: $15 + 23$.
First, think of $15$ as $10 + 5$, and $23$ as $20 + 3$, so you have $$15 + 23 = (10 + 20) + (5 + 3) = 30 + 8 = 38$$
Granted, it means that you have more numbers to add, but when those numbers are "easy" for you to add, it's not too bad. The more you practice this, you'll soon be able to add numbers "on the spot" more directly.
Note: see ShreevatsaR's comment right below this post. It's pointed out there that in situations like the example I use above, one can simply break apart, say, $23: 23 = 20 + 3$; then add $15 + 20 = 35$; and finally, add $3 + 35 = 38$. Chances are, you likely already do some of these mental "calisthenics" without considering it an arithmetic problem.
Perhaps it would help us to know what hasn't worked for you, or if you have trouble only when you have to do so "on the spot", without paper and pencil (or calculator).
The biggest suggestion I have is for you change your attitude about the task of adding from approaching it with dread -- or by simply avoiding it! -- ("Oh no, adding! I can't add!"), to approaching it as a challenge. ("I can do this, and I will!")
Another helpful resource is the Math Page, where there is a sequence of lessons to help develop skills in arithmetic, in general (again, free).