Hi and welcome to the forum
Get a 10 gallon tank or 20 gallon tall tank. They have much more space for plants and only cost a few dollars more. They also take up a similar amount of space to the smaller tanks.
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LIGHTING FOR PLANTS
Most aquarium plants like a bit of light and if you only have the light on for a couple of hours a day, they struggle. If the light doesn't have a high enough wattage they also struggle. Try having the tank lights on for 10-12 hours a day.
If you get lots of green algae then reduce the light by an hour a day and monitor the algae over the next 2 weeks.
If you don't get any green algae on the glass then increase the lighting period by an hour and monitor it.
If you get a small amount of algae then the lighting time is about right.
Some plants will close their leaves up when they have had sufficient light. Ambulia, Hygrophilas and a few others close their top set of leaves first, then the next set and so on down the stem. When you see this happening, wait an hour after the leaves have closed up against the stem and then turn lights off.
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PLANTS
Some good plants to try include Ambulia, Hygrophila polysperma, H. ruba/ rubra, Elodia (during summer, but don't buy it in winter because it falls apart), Hydrilla, common Amazon sword plant, narrow Vallis, Water Sprite (Ceratopteris thalictroides/ cornuta).
The Water Sprite normally floats on the surface but can also be planted in the substrate. The other plants should be planted in the gravel.
If you add an iron based aquarium plant fertiliser, it will help most aquarium plants do well. The liquid iron based aquarium plant fertilisers tend to be better than the tablet forms, although you can push the tablets under the roots of plants and that works well.
I use Sera Florena liquid plant fertiliser but there are other brands too.
Do not bother adding carbon fertiliser to the tank because they are not necessary.
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LIGHTS
Stress from tank lights coming on when the room is dark can be an issue. Fish don't have eyelids and don't tolerate going from complete dark to bright light (or vice versa) instantly.
In the morning open the curtains or turn the room light on at least 30 minutes (or more) before turning the tank light on. This will reduce the stress on the fish and they won't go from a dark tank to a bright tank instantly.
At night turn the room light on and then turn the tank light off. Wait at least 30 minutes (or more) before turning the room light out. This allows the fish to settle down for the night instead of going from a brightly lit tank to complete darkness instantly.
Try to have the lights on at the same time each day.
