I’m currently travelling through Italy and I can have full strength 4G but almost no data connectivity. I don’t know how anyone manages to use their data allowance here.
Have you got a new SIM from a local operator (which one?) or are you using your old one ? In the last case perhaps there is a problem in the roaming config.
In Italy the connection is pretty good (50 down /10 up Mbps), at least in the urban areas.
Basic contracts give you unlimited voice, unlimited texts, 50-70GB/month for about €8.
Depends on where you are of course - a friend of mine sent me a speedtest screenshot amused at how his download speed on a mobile connection in northern Italy was ten times faster (literally) than his ADSL connection in south London!