Lights dimming and brightens and slow start to push button

Welcome to 2CarPros.

I have to be honest, it sounds like the alternator is going bad. The dimming lights and slowing fan indicate to me that the power supply is going up and down. If the battery is good, the battery terminals are clean and tight, and the main ground to the engine from the battery is clean and tight, then suspect either the alternator belt is loose or the alternator is going bad.

It's somewhat easy to check an alternator. All that is needed is a volt meter or a multi meter. Once the meter is set to DC voltage, check the battery voltage with the engine off. It should be slightly over 12v. Then, start the car. It should go near 14v at the battery if the alternator is working properly.

Here is a link that explains how this is done. I have a feeling that if you check it when experiencing the problem, you will see the voltage vary.

https://www.2carpros.com/articles/how-to-check-a-car-alternator

https://www.2carpros.com/articles/how-to-use-a-voltmeter

https://www.2carpros.com/articles/how-to-check-wiring

If you find the alternator is an issue, here is a link that shows in general how it is replaced:

https://www.2carpros.com/articles/how-to-replace-an-alternator

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Here are the directions specific to your vehicle for alternator replacement. The attached pics correlate with the directions.

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2016 Jeep Truck Renegade 4WD L4-2.4L
GENERATOR - REMOVAL AND INSTALLATION (2.4L)

REMOVAL
Disconnect and isolate the negative battery cable. If equipped with an Intelligent Battery Sensor (IBS), disconnect the IBS connector first before disconnecting the negative battery cable.
Remove the engine cover. (Refer to 09 - Engine/COVER, Engine/Removal and Installation) .
Remove the right front wheel. (Refer to 22 - Tires and Wheels/Removal and Installation) .
Remove the accessory drive serpentine belt. (Refer to 07 - Cooling/Accessory Drive/BELT, Serpentine/Removal and Installation) .

Pic 1

Disconnect the upstream oxygen sensor wire harness connector (1).

Pic 2

Disconnect the oxygen sensor wire harness and remove the bracket (1).
Remove the exhaust manifold upper heat shield bolts (2) and then remove the shield.

pic 3

Remove the nut and disconnect the B+ electrical connection (1) from the generator.

pic 4

Remove the upper and lower generator mounting bolts (1).
Remove the generator from the engine compartment.

INSTALLATION
Position the generator for installation.

pic 5

Install the upper and lower generator mounting bolts (1). Tighten the bolts to the proper (Torque Specifications) .

pic 6

Connect the B+ electrical connection to the generator stud, then install and tighten the nut to the proper (Torque Specifications) .

pic 7

Install the exhaust manifold heat shield (2).
Install the oxygen sensor wire harness bracket (1).
Position and secure the oxygen sensor wire harness (1).

pic 8

Connect the upstream oxygen sensor wire harness connector to the oxygen sensor (1).
Install the accessory drive serpentine belt. (Refer to 07 - Cooling/Accessory Drive/BELT, Serpentine/Removal and Installation) .
Install the right front wheel. (Refer to 22 - Tires and Wheels/Removal and Installation) .
Install the engine cover. (Refer to 09 - Engine/COVER, Engine/Removal and Installation) .
Connect the negative battery cable. If equipped with an Intelligent Battery Sensor (IBS), connect the IBS connector.
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Before just replacing the alternator, in addition to the aforementioned test, I would remove the alternator and have it bench tested at a parts store. Most will do it for free. However, the link above will help guide you and let you know if there is an issue.

Let me know if this helps or if you have other questions.

Take care,
Joe