In your sample input file, it is the first field and only the first field, that is quoted. If this is true in general, then consider the following as a method for deleting the second and third columns:
$ awk -F, '{for (i=1;i<=NF;i++){printf "%s%s",(i>1)?",":"",$i; if ($i ~ /"$/)i=i+2};print""}' file
"abc@xyz.com,www.example.com",field4
"def@xyz.com",field4
As mentioned in the comments, awk does not natively understand quoted separators. This solution works around that by looking for the first field that ends with a quote. It then skips the two fields that follow.
The Details
for (i=1;i<=NF;i++)
This starts a for over each field i.
printf "%s%s",(i>1)?",":"",$i
This prints field i. If it is not the first field, the field is preceded by a comma.
if ($i ~ /"$/)i=i+2
If the current field ends with a double-quote, this then increments the field counter by 2. This is how we skip over fields 2 and 3.
print""
After we are done with the for loop, this prints a newline.
