You could try something like the following to allow traffic to specific websites and then deny all other web/https traffic. Keep in mind that in the configuration I am assuming you have two seperate interfaces for the two network connected to the ASA (that could be two subinterfaces or two seperate physical interfaces.
access-list LAN1 permit tcp 192.168.2.0 255.255.255.0 any eq http
access-list LAN1 permit tcp 192.168.2.0 255.255.255.0 any eq https
regex domainlist1 "\.yahoo\.com"
regex domainlist2 "\.myspace\.com"
regex domainlist3 "\.youtube\.com"
class-map LAN
match access-list LAN1
class-map type regex match-any DomainList
match regex domainlist1
match regex domainlist2
match regex domainlist3
class-map type inspect http match-all DomainClass
match request header host regex class DomainList
policy-map type inspect http http_inspect_policy
class DomainClass
inspect
class class-default
drop
policy-map inside1-policy
class LAN1
inspect http http_inspect_policy
service-policy inside1-policy interface inside1
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