Dificulty in starting. The engine turns over,starts then cuts out. This is getting progressively worse - hot or cold

Do you have a check engine light on? Have you scanned for fault codes?

forename?
07 forenza (seems all posts are for this 1 suzuki)
a problematic car for sure....?
in what country, car has 6 engines, but 1 in USA>
so cant help you lacking where you are, (name a country,not your IPaddress)
we have the 2.0L engine, in USA< this is a USA forum so......??

ok the computers in your car are way smarter than say 1988 Samurai.
the PCM that runs the engine, for SURE cuts AC to keep engine from self destructing!! AC can do that,,,,,,
want a cracked or warped engine head ?
no.
the PCM cuts the a/c, and if engine is overloaded. or overheated
so forget the A/C
fix engine overheating (primary issues)
and guess wha?t , that gets worse, so delaying the fix
a $5 part compounds to $3000 new engine.
why not just get car fixed.???
"we cant teach you 50 years of engine problems in 5min time."
best post ever that.. here.

1: fuses good? check them it means.
2: water leaks?
3: levels wrong? (in actual rad, cold tested)
4: fan belts loose?
5: bad thermostat
6: bad water pump, impeller rusted to zip, do to abuse.
7: over heats parked only or moving fast or only up hills,? key facts if only parked the fan is dead.
8; lots more, but i end here

you need to check the engine system.Maybe obd2 code reader can help you. OBD2 Code Scanner

intermittent are hard to find.
as they say" if it ain't broke, you can't fix it" applies
but in truth, we lay in traps to find this.
this is true with all machines, not just sx4
facts: (random no starts) but always cranks fast.
only fails cold engine. never hot. (does restarts hot work?)
seems yes.
seems you have full 12vdc power at all times.
seems the engine is ok. as it is not heat rlelated or the classic hot soak fails.
so what is it.
well, its 1 of these.
1: lost spark, simple tests, do so when failed.
2: bad fueling. (too much or too little)
the tests for each failure are dead simple and work on any engine.
ask.

if your battery, and starter and plugs and wires is ok, then i would give the vehicle a fuel pressure test to see if the fuel pressure is ok according to the specs, if it is ok, i would check the fuel injectors.

overheating starter possibly. If possible find a way to have a system test done at a parts store.

You may have created an over charged condition and at this point the a/c compressor will cut off on high head pressures. This can be verified by installing gauges to the systems high and low side service taps and reading the pressures.An overcharged system will start the compressor briefly but soon cuts off once pressure goes to 350 - 400 lbs on the high side. Acceptable pressures for this system should be no greater than 250 high and low suction should drop to 40-50 psi.You might recover the charge and evacuate the system,then try bringing the charge back to proper level.A sight glass in the reciever drier really helps determining proper fill.Once all bubbles have dissipated from the sight glass during charging ,one should stop at that point to avoid overfilling.Too much freon is worse than a low charge especially in hot ambient conditions....

it starts from cold because the auto choke is on ,allowing a richer mixture ,try removing the sender unit in the tank and seeing if the connection from the pump to the outlet connection is ok or leaking causing a loss of fuel pressure

• Assumed your AC unit is OK, but uneven temperature distribution. Try to throw the heater on an off severtime. A sticky valve on the heater would cause similar problem.
  
• Observe the RPM while parking your car. While the AC cut in cut out (or turn on an off). If it cause no load on the engine. The freon may be low or the compressor is bad. In this case only the intake side of the evaporator is cooler than the other side.
  

I had a similar problem with my 1986 300e , which got progressively worse until the car was no longer driveable, I changed out the fuel pump relay and the problem went away.