Green tea is not to everyone's taste. You can try adding honey instead of sugar. As honey has more of a distinct flavour than just sugar it may help mask the green tea flavour more successfully. Another suggestion is to add some mint leaves if you're a fan of mint. If you're drinking green tea solely for health reasons the mint can be excellent for aiding digestion.
Certain tea companies also sell 'light' green tea. This can be more palatable, especially for people like yourself who don't like the taste of steeped green tea so changing your brand of tea might be helpful.
It may be that with perseverance you may acquire a taste for green tea (I personally used to hate camomile tea but continued to drink it every day and now it's my favourite).
Hojicha is a green tea which is made from bancha, a low grade green tea, and cooked slightly; this very inexpensive green tea often comes out brown because it is discolored by oxidation. Other than this variety, and some very stale bancha, I can't think of a Japanese green tea that comes out brown. Some stale kukicha might come out brown, and low quality genmaicha made with poor quality kukicha could be brownish from the combination of excess oxidation and the toasted rice.
Most of the non-Japanese green teas I've seen sold as "green tea" and some lower quality teabags marked as "sencha" are broken down tea leaves that can easily oxidize in the packaging to the point where they are, at best, yellowish.
However, if you use even a moderate quality sencha, and it isn't stale, it should come out green. Most good quality kukicha and genmaicha will at least come out greenish-yellow. Since most Japanese restaurants use a fairly inexpensive second-harvest sencha, I imagine you're either buying hojicha or simply using a very stale, low-quality sencha.
Chinese green teas are sometimes slightly yellow, but if they're actually yielding a brown brew, they, too are either stale or are simply mislabeled oolong.
I used to sell a fair amount of green tea when I ran a small scale import business. I do recall even matcha products turning brown after being exposed to excessive sunlight; one of the bakeries I know was constantly running into issues with their matcha pound cake slices on sunny days, especially those pieces in the front of their pastry case. So it's reasonably possible you simply have tea which has been stored improperly; you're best off with a nitrogen flushed tea and it should be used within about 6 months of opening the sealed package. If your tea comes unsealed, it probably isn't in very good shape to begin with.
In any event, my recommendation is to try a first-harvest high-mountain grown sencha that is sold in nitrogen-flushed aluminum packages. Depending on the style, the brewed color will be either very green or slightly yellowish if brewed in the typical way. You can either do a very brief, several second infusion at near-boiling, or a longer infusion at around 80 celsius. I like it both ways.
Gyokuro, if you can afford it, is extraordinarily green, but is unlikely to be served at a restaurant without an additional charge (it's essentially the same leaves used to make matcha, except rolled instead of being milled).
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Have you tried using one green tea bag and one bag of herbal tea? There are many herbal tea sampler packs that you could get to try several flavors of tea. Since you are making iced tea, I would also recommend adding chopped fruit once the tea has chilled - I would imagine some crisp apples or even a fresh melon would add a really refreshing profile. You can even let the fruits steep in the chilled liquid, like a tea-sangria.
Another option is to create a flavored simple syrup in order to sweeten your iced tea. It's always better to use a simple syrup in a cold liquid than any kind of 'dry' sugar. Simple syrup is easy to make on the stove or in the microwave, and just add some flavors you might like while heating it. I wouldn't recommend vanilla for green tea (gut reaction), but maybe lavender, orange or cardamom?