I hope that 70 kg is a typo. 70 kg = about 154 pounds! I pasted some info below, but you need to go to the link to verify exactly what you have.
Take another look at Norwegian's Baggage info. Here's the link to it for both checked and carry-on. And below is a quote from the Checked Baggage section. Perhaps you were confusing the max for 2 checked bags. Or more likely you meant 70 pounds instead of kilos.
But I think you are confusing the actual free limits with the maximum you can take and pay a fee.
"International long haul flights
No bags included
1 x 20 kg
Flex, Premium and PremiumFlex
2 x 20 kg"
Excess baggage charges apply to any items over the 2 baggage limit, and to items over 20 kg – 11€ per kilo, per leg."
"Size and weight limits
Pay careful attention to the size and weight of your bags when preparing for your trip.
Each bag must not be heavier than 32 kg or lighter than 2 kg.
Your total amount of checked bags cannot weigh more than 64 kg. Each bag must not exceed 250 x 79 x 112 cm (L x H x W) with max. circumference 300 cm.
Now for the Hand Baggage.
"One carry-on bag included
You can always bring one carry-on bag in the cabin free of charge. You can also bring one small personal item on board. This could be a little handbag or a slim laptop case that fits comfortably under the seat in front of you. Your ticket type will determine how heavy your hand baggage can be.
Size and weight limits
LowFare, Lowfare+, Premium
1 carry-on bag: max. 55x40x23 cm
1 small personal item: max. 25x33x20 cm
Max. combined weight: 10 kg (about 22 pounds)
1 carry-on bag: max. 55x40x23 cm (about 21.5x15.5x9 inches. That's including wheels and handles.)
1 small personal item: max. 25x33x20 cm (about 9.8x 13x7.8 inches.)
Max. combined weight: 15 kg (about 33 pounds)"
As for the sizing and weight, I'd assume they will check. I flew British Airways a couple of years ago and downsized my personal item because of that, but their size was more generous (40x30x15 cm) than Norwegian's appears to be.
The Norwegian sizes are the same regardless of ticket type, but the combined weight is not, and yes, BA did have me put both my carry-on and my personal item on the scale at the same time. I'd expect Norwegian to do the same. The numbers in parenthesis are mine.