In European Spanish, people pronounce the combinations:
za ce ci zo zu
As "th" in "thanks".
Tha the thi tho thu.
In American Spanish, this are pronounced as "s"
sa se si so su.
You can pronounce them either way and be correct. It is like the pronunciation of "t" after "n" In American English and European English. In American English the "t" is not pronounced, in European English, the "t" is pronounced. Since languages change and they develop different features if members of one group, speaking one language, are separated for several generations, it is normal that they develop different pronunciations. None of them can be said to be wrong. They are just different versions of the same language. Spanish speaking people from America and Europe will understand you in either case. Just pick one up and use it consistently.