In addition to Adam Davis's correct answer, I would like to post my own solution for this operation.
As list is something big, there is three of many differents tested solutions...
First prepare your TLD List in that way:
wget -O - https://publicsuffix.org/list/public_suffix_list.dat |
    grep '^[^/]' |
    tac > tld-list.txt
Note: tac will reverse list to ensure testing .co.uk before .uk.
posix shell version
splitDom() {
    local tld
    while read tld;do
        [ -z "${1##*.$tld}" ] &&
            printf "%s : %s\n" $tld ${1%.$tld} && return
    done <tld-list.txt
}
Tests:
splitDom super.duper.domain.co.uk
co.uk : super.duper.domain
splitDom super.duper.domain.com
com : super.duper.domain
In order to reduce forks (avoid myvar=$(function..) syntax), I prefer to set variables instead of dump output to stdout, in bash functions:
tlds=($(<tld-list.txt))
splitDom() {
    local tld
    local -n result=${2:-domsplit}
    for tld in ${tlds[@]};do
        [ -z "${1##*.$tld}" ] &&
            result=($tld ${1%.$tld}) && return
    done
}
Then:
splitDom super.duper.domain.co.uk myvar
declare -p myvar
declare -a myvar=([0]="co.uk" [1]="super.duper.domain")
splitDom super.duper.domain.com
declare -p domsplit
declare -a domsplit=([0]="com" [1]="super.duper.domain")
Quicker bash version:
With same preparation, then:
declare -A TLDS='()'
while read tld ;do
    if [ "${tld##*.}" = "$tld" ];then
        TLDS[${tld##*.}]+="$tld"
      else
        TLDS[${tld##*.}]+="$tld|"
    fi
done <tld-list.txt
This step is a significantly slower, but splitDom function will become a lot quicker:
shopt -s extglob 
splitDom() {
    local domsub=${1%%.*(${TLDS[${1##*.}]%\|})}
    local -n result=${2:-domsplit}
    result=(${1#$domsub.} $domsub)
}
Tests on my raspberry-pi:
Both bash scripts was tested with:
for dom in dom.sub.example.{,{co,adm,com}.}{com,ac,de,uk};do
    splitDom $dom myvar
    printf "%-40s %-12s %s\n" $dom ${myvar[@]}
done
posix version was tested with a detailed for loop, but
All test script produce same output:
dom.sub.example.com                      com          dom.sub.example
dom.sub.example.ac                       ac           dom.sub.example
dom.sub.example.de                       de           dom.sub.example
dom.sub.example.uk                       uk           dom.sub.example
dom.sub.example.co.com                   co.com       dom.sub.example
dom.sub.example.co.ac                    ac           dom.sub.example.co
dom.sub.example.co.de                    de           dom.sub.example.co
dom.sub.example.co.uk                    co.uk        dom.sub.example
dom.sub.example.adm.com                  com          dom.sub.example.adm
dom.sub.example.adm.ac                   ac           dom.sub.example.adm
dom.sub.example.adm.de                   de           dom.sub.example.adm
dom.sub.example.adm.uk                   uk           dom.sub.example.adm
dom.sub.example.com.com                  com          dom.sub.example.com
dom.sub.example.com.ac                   com.ac       dom.sub.example
dom.sub.example.com.de                   com.de       dom.sub.example
dom.sub.example.com.uk                   uk           dom.sub.example.com
Full script containing file read and splitDom loop take ~2m with posix version,  ~1m29s with first bash script based on $tlds array, but ~22s with last bash script based on $TLDS associative array.
                Posix version     $tldS (array)      $TLDS (associative array)
File read   :       0.04164          0.55507           18.65262
Split loop  :     114.34360         88.33438            3.38366
Total       :     114.34360         88.88945           22.03628
So if populating associative array is a stonger job, splitDom function become lot quicker!