From what I have heard, the Polish "sz" is actually closest to the German "sch." The "š" to me sounds more like the Polish "ś" and that is approximated in English with the "sh" diagraph. The "sz" is a rougher sound, the "ś" is soft.
So, to answer your question…
Jaszowiec = Yah-sho-vyetz (with English phonetics), but for the "sh" you have to move your tongue forward to get the proper sound.
I hope that y explanation isn't too confusing… lol.