As you already figured, the baggage information on Turkish Airlines states a basic 20kg per passenger for international flights, with a lot of exceptions.
Looking at the baggage information page, I can see exceptions for flights from/to USA and Canada, South America, many African countries and some European countries like Germany and Ireland. There are even exceptions based on the season. There is however no exception for UK. But the most important part of the page is maybe the message at the very top:
The free cabin and checked baggage allowance may vary according to reservation class and departure runway. Please check your ticket for information regarding your free baggage right.
For interline flights, the baggage policy of the airline designated Most Significant Carrier by the system will be in effect.
So what matters is not so much the (complex) content of that page, but the ticket. You should read your ticket (if you received multiple emails from your travel agent, this is usually written at the beginning of the email if it is or not a ticket) and only this condition applies.
Also, even though Turkish Airlines is most likely operating both of your flight segments, there is also the rule of the Most Significant Carrier. This means what applies is the allowance of the airline operating your longest/most significant segment:
If another airline is involved in your whole journey; the most significant carrier’s (MSC) free baggage allowance rule will be applied. MSC is assigned according to which airline operates the most significant and/or longest part of the journey.
On the baggage page of Turkish Airlines, there is also a distinction between some cases giving you 40kg of allowance or 2 pieces of 20kg. So if this is the case, you may bring one bag of 40kg. However, as @AndrewFerrier commented, most airlines usually forbid bags over 32kg, most likely to respect the back of airport workers. This is actually probably a request from airports. So, for their health, and to avoid paying overweight fees, it is probably better to bring bags weighing less than 32kg.
So in the end, you should look at your ticket to know your baggage allowance.