Do I need to be concerned if the engine temperature fluctuates?

Hello,

The temperature gauge should not go much above the middle or the normal range.
If it is above the middle and close to high, then the engine is running hotter than it should.
Which could be due to any number of reasons. Could be due to having low coolant, faulty water pump, cooling fan not working properly etc.

Take a look at this guide:

https://www.2carpros.com/articles/symptoms-of-an-overheating-engine

Let me know of any questions.
Thank you.

Hey SQM, I have given to a mechanic at Bridgestone, we have checked for leakages, thermostat, water pump, coolant has been changed, radiator fan as well but still that problem has not been solved.

If all those checks out to be good, but the gauge still shows high, it is possible that the gauge could be incorrect, which can be due to a bad temperature sensor.
Did they check to see if the engine actually running hot?

Hey SQM, the gauge goes up and comes down, I have noticed that when I turn left, the gauge comes down to the middle, and while turning right or going uphill it goes up.
When the car is stopped the gauge is at the middle, so how do I check if the engine is going to hot or not.

The easiest way to check the temperature would be to use an inferred thermometer to check the temperature of the upper radiator hose when the engine is at operating temperature.
This method won't be pinpoint accurate, but it will provide you with a good understanding of the engine temperature.
The operating temperature should be around 174F to 194F.

Alright, but I have noticed that the temperature gauge comes back to center when I turn left. So, I am not understanding what the problem is, as it goes up for a second, not to the H, but a line below the H line, you can see in the image. It just fluctuates between them.

It is possible that you might have air in the coolant system, maybe a air pocket trapped close to the sensor, and that could be triggering the fluctuating reading.
Try this, when the engine is cold, remove the coolant reservoir cap. Then start the engine.
Now carefully squeeze the upper radiator hose several times and watch for any bubbles out of the reservoir. Do this for a few minutes as the engine warms up.
Be sure to top of coolant if necessary.
Then drive as normal (after putting back the cap).
Let me know what happens.

Hey SQM, I saw this, when I turned down my car. It is the lower hose, I think.

Is it leaking out of that hose?

I saw smoke, so I opened bonnet and saw that, still not sure if that is it though.

Typically, when coolant leaks it will get on the hot surface of the engine or exhaust and that will create smoke.
Try starting up with the hood open and see if it is leaking out of the hose. If it is, then that will have to be replaced.

It is not leaking from there, mechanic told me it could be the head gasket.

A blown head gasket will create white smoke out of the tailpipe. Also it can make the oil a milky color.
Take a look at this guide of what to look in the event of a possible blown headgasket.

https://www.2carpros.com/articles/head-gasket-blown-test

Let me know what you find.