A Boeing 737 (average sized aeroplane) has a take-off capacity of around 80,000kg.
Once you account for the aeroplane itself, the max fuel load it can carry, and the crew it has around 24000kg of load to spare.
The average number of people one of those is fitted out to carry is 150. meaning each passenger has an allowance of 133.3kg.
The average weight of an adult in the UK is 78.5 kg and average allowance for checked baggage is 23kg, so a per passenger total of 101.5 kg (but let's say it's closer to 110.5 kg to allow for clothing and hand luggage).
That means that, even if a 737 was exclusively filled with adults who've all checked a suitcase and carried on some hand luggage, there would still be around 7425 kg of spare capacity.
There's no issue with the weight capability of commercial aeroplanes.