GT350R wroteI have to disagree with most people on this one, I have owned a Tiburon GT but mine was '03. Now mine lasted 2 years with a turbo, however, I did build the motor to be able to handle it. It didn't have 60 psi, I was running right at 9-12 pounds. If you want to turbo this car, you have to sink some cash into it first. Which, will end up costing you more than just getting a different car in the end.
Of course the Tiburon can be turbo'd and can be fast. I just don't know why you would want to start off with a slow vehicle opposed to something faster.
You did not specify how much power you made with you car turbo'd and how much you spent on it. I seen a few Youtube videos about turbo'd Tiburon's and they made any where from 220 - 350 hp. Keep in mind that they only make around 170 stock. And stock they weigh around 3050 pounds.
Stock they will make approximately a 14.8 second pass in the quarter mile. Now lets say that you turbo the car and it makes around 300 hp and weighs around 3175. Then you are making about a 12.5 second pass. That turbo kit is going to add around 100+ pounds. You can buy a cheap universal turbo kit on ebay for ~1000$ but it is going to need some modifications and will probably end up costing more money. Then to be able to make 300+ hp you are going to need to build the motor. And that is being generous. Normally a turbo will add around 50% to 65% power to a vehicle.
KBB prices the Tiburon at around $4000 for a stock good condition car. So add $1500 for turbo kit and another $300-500 to build the motor a little. That comes out to be around $5800. That is alot of money for a relatively slow vehicle. And in my opinion nothing sounds as good as a v8.
Or you could buy something like a 2002 Camaro with a ls1. Stock these make around 325 hp and weigh around 3500 pounds. In a quarter mile the Camaro should run around 12.6 seconds stock.
KBB prices the Camaro anywhere from $4600 - 6800 USD. So for the same amount of money or less you could have a vehicle that runs relatively the same as the tiburon and have a much better base to build off of. Swap a cam and put some headers on for around $500 - 1000 USD and add around 75 to 100 hp after a tune. So now you have a car that weighs about 3500 and makes around 400 hp. In a quater mile you would make about a 11.7 second pass while still being NA and mostly stock.
Just as a side note, I don't know much about tiburons and turboing them. I am just going off of what a ls turbo build would be.
And this is all assuming that you are doing the work yourself. If you were to have a shop do the things listed above, it would cost even more.
OP, if you like the Tiburon then keep it and build it. I personally would start with a different vehicle, but it is not my car so my opinion should not matter that much.