Here's more details on what parts and why your sentences don't sound right.
1. I don't agree with the way to educate children as the picture shows.
First, "shown/depicted in the picture" is clearer than "as the picture shows" in this case ("as the picture shows" is best when you start off an independent idea in the main clause, like "As the picture shows, I/We/They ...", ).
Second, the to-infinitive in "agree with the way to ..." is probably not the best choice. When referring to an existing or factual thing, "the way/method of VERB-ing" or "the way that (... clause)" is usually better (roughly speaking, "of V-ing" = objective, existing, factual; "to V" = subjective, what's to come, probable).
So you might say:
- I don't agree with their way of educating children shown in the picture.
- I don't agree with the way they are educating children as depicted in the picture.
- I don't support/like the way they are educating children in the picture.
2. The phenomenon like the picture exists commonly.
"like NOUN" is best used when NOUN is an example of the same kind as the antecedent, as in "things like pictures", or "people like you", Here, you're not using the picture as an example of a phenomenon but referring to what it depicts, so a more elaborate phrase like "shown/depicted in" is in order.
Also, "exists commonly" doesn't seem the most natural. "is common" or "commonly occurs" is probably better.
So I would recommend:
- The phenomenon depicted in the picture is common.
- The phenomenon depicted in the picture commonly occurs.
- What's shown in the picture is a commonly occurring phenomenon.