I am traveling with TAM from GRU to LHR then Connecting to IST with TK what is better to let them tag my luggage to IST or only to LHR and move my luggage my self to the TK flight
Thanks

If you tag only to LHR you need to go thru immigration/customs and recheck.
If you are running short of time you possibly miss your connecting flight.
If your luggage does not show up in LHR you need to file a lost report there and also risk to miss the connecting flight.

What would be the advantage of tagging only to LHR?

I have about 3 hours for my next flight
Tag to LHR so I make sure TAM does not lose my bag before it gets to TK
I don't know how it works there or do they do a good job in LHR with connecting bags

If possible in GRU, check your luggage through to IST and just enjoy transferring at LHR, there are a lot of shops/bars/restaurants/lounges on the airside. AFAIK TAM and TK are both *A. Does not mean they use the same groundhandling system, but *A should be in your advantage.

If you decide to collect your luggage after the GRU flight and check it in for TK then 3h in LHR can be hit and miss. You could have an inbound delay of 20-30min, then wait another 15-20min because your position is still occupied = cruising around on the tarmac. In general LHR ways in the Terminals are long. At customs, if you are a non european the queue can be 1h or more. LHR has mainly widebody aircraft and therefore a lot of pax everywhere. Then find your luggage. Then for checking in again you would need to queue up again with TK (or *A checkin) and then security again etc...and then you mostlekily have missed your flight or had a stressfull time in LHR.

But it could all work out as well, anyway, report back how you did manage.