Learn English – start a sentence with “Besides” just followed by a comma

Run to me, I get you to some place to hide.

This sentence starts off just fine. Verbs that command to someone to do something have the "implied you" as the subject. This is called an imperative sentence. As it says at one website:

IMPERATIVE
An imperative sentence gives a command. For example: "Shut up and kiss me."
Note that an imperative sentence does not require a subject; the pronoun you is implied.

That said, the sentence could still be improved; "someplace" is one word, and the second half of the sentence should include the word "will":

Run to me, I'll get you to someplace to hide.

Observed the experiments, I learned that "Failure is a mother of Success".

This sentence has problems. The main verb here is learned, not observed. You could fix this in one of two ways:

Fix No. 1: Start the sentence with a prepostional phrase

This would require you to change the word "observed" to its gerund form:

By observing the experiments, I learned that "Failure is a mother of Success".

Fix No. 2: Start the sentence with the subject and use a conjunction

This could be accomplished by using a compound verb:

I observed the experiments and learned that "Failure is a mother of Success".

or by making it a compound sentence:

I observed the experiments, and I learned that "Failure is a mother of Success".

This sentence is both correct and makes perfect sense.

"The water had overflowed" is, on the face of it, fine. Overflowed means to spill beyond capacity, and that's definitely a thing water sometimes does.

It's often followed by a description of what was overflowed, but there's no rule that says that has to be the case. Instead, here, the prepositional phrase "under the bridge" describes where the overflow happened.

Since it is rather normal for water to be under bridges, that's not at all surprising. More context would make it completely clear, but without that, I assume that a river or stream overflowed its banks below a high bridge.

Like this:

Or, alternately, it could be a bridge over a road, and water may have overflowed onto that road (or, using a slightly alternate meaning of "overflowed", simply overflowed the road). Like this: