Electronic – Using low-voltage halogen transformer for LED

Too low resistance of the bulbs causes too high current in the transformer secondary winding and so the transformer gets overheated. If you short the transformer (extreme of what you have with a too powerful load) the transformer gets burnt even faster (don't try that).

Added:

You advise 3W load on a 50 W transformer.
I suggested 85-90% efficiency.
Transformer will draw magnetising current when unloaded. This will incur copper losses but will not be the full loaded current.
10W additional = 80% of rated power which sounds much too high.

You have still declined to tell us HOW you are measuring power - you say it jumps by 50 Watts. Measured how & with what. The only certain measure would be a power factor correct Watt meter. ie the house meter would qualify - a plug in watt meter quite possibly wouldn't.

More information is needed.
You mention "halogen mal transformers" - are they 50 Hz iron core transformers or smps (switch mode power supply) converters?

You imply that the LEDs operate from 12V and
imply that they may usually run on DC.
You suggest that they run OK on the 12 VAC from the "transformers" which may be
50 Hz AC or
10's to 100's of KHz AC?

You say "it would appear that you are using 15W/LED lap but knowing how you measure this in as much detail as possible could be useful. Or not :-).
You imply that it is not just the VAC x IAC product as yoyt say that in the output VAC x IAC = 15W but compensating for RMS gives 5W. This raises the question as to what it is you measure and how to get the 15W and how you RMS convert to get 5W. (The factor of 3 does not untuitively drop out of any conversion that comes to mind).

SO if you can fill in all the bgaps, and add a diagram if your word picrure would benefit from it, it may help.

Iron core power supplies may be in the 85% - 90% efficiency range but better is conceivable, and any fool can get less than that, and some do. (A 60Hz designed transformer run at 50 Hz can get nice and toasty. Ask me how I know :-).)

An electronic "transformer" can get just about any efficiency at all. 95%+ is doable and 65% would not be a surprise in some few cases. 80%-90% should be hoped for if not always expected.

Running LED lamps that are "designed" [tm] for DC use on AC may cause problems. At 50 Hz flicker may be an issue. IF the lamps have intelligent and/or active internal regulation to a lower DC voltage or to constant current then conceivably the need to provide a pulse of power input twice per cycle (or quite possible once per cycle if they are polarity dependant ) may cause massive converter overload and may even cause core saturation problems upstream in the 230:12 VAC supply. For extra points they may have placed a reverse diode across the supply input for "protection" and this may be causing a massive current peak every unused reverse half cycle.

SO too many guesses needed, not enough data.
A much fuller description should help.

IF you take the 12 VAC, feed through a full wave bridge and then add a LARGE electrolytic cap so there is well under 1 Volt ripple on the DC, and drive the lamp from that, what power does it draw?

Any web links for transformers, LED lamps, ...?