Learn English – the difference between “What happened?” and “What has happened?”

No, that is not correct.

I have broken my leg (without the yesterday) can be used in two situations that I can think of:

1) "I have broken my leg four times in the last year!"

In this case it means that your leg was broken at four different times in the past year, but it doesn't necessarily say your leg is broken now. I might be broken now or it might not; it doesn't imply anything either way.

2) "I've broken my leg!"

I can see this being said as a surprised exclamation at the moment your leg is broken. As in, "Oh my god, my leg just broke!" This means that, at this moment, you have just broken your leg. Now whether or not anyone is coherent enough to actually say this when they break their leg is another matter, but it would be valid at the time if they did say it.

Now, if you broke your leg yesterday, you cannot accurately say "I have broken my leg." Your leg is currently broken, yes, but we say:

I broke my leg yesterday.

The action of breaking occurred yesterday, so you use the past when talking about it. If you want a sentence in the present, you can say something like this:

My leg is broken; I had an accident yesterday.

1. Present perfect cannot be used with a temporal expression which does not include the present. The present perfect cannot be used in this sentence because it locates the action entirely in the past: yesterday. You must use the simple past.

2. In this case ‹read› is not understood as simple present but as simple past. The two tenses are pronounced differently (present: /ri:d/, past: /rɛd/), but have exactly the same form in writing. Consequently, ‹read› is correct here.

3. It is only occasionally true that the present perfect "specifies an action that has recently finished". The present perfect has other uses, too.

For a detailed introduction to the uses of perfect constructions, see What is the perfect, and how should I use it?, especially §§ 3.1 Grammatical meaning, 3.2 Pragmatic meaning and 4. When and how should I use the perfect?.