Post if you can, saves every few turns, so that we can have a clear idea of how you are making the game approach. Or, explain as best as you can.
What kind of map are you playing, what Pantheon did you choose, how many cities do you have on turn 100 etc. Scientific victory with Gilgamesh is quite straightforward;
Expand as much as you can until you are touching borders with other AIs (you can go to war too, but if your enemy is too far, you will have loyalty issues and the war could cost more than what you expected to get from it). Use your builder charges to improve the resources, and build as many Ziggurats as you can in your empire. Preferably by the river, so you can get the culture. Build your farms in other places. Then you can start building infrastructure in a sort of hierarchy/preference method. Break the Ziggurats when you can place a district on top if its going to be a better alternative for that tile, like aqueducts, industrial zones or whatever you need. They are a temporary boost for your early game science and culture while you expand, and they do a very good job at that, allowing you to delay Campuses and Theatre Squares a lot while still keeping up/catching up.
Clear all the barb camps you can, Gilgamesh gains extra benefit from them.
Other than that, just keep trying, and dont feel bad because you see yourself still behind in turn 100 (I know, easy to say, my anxiety levels go through the roof too if I think Im doing fine and suddenly I meet a civ in turn 150 that is doubling me and every other AI in every aspect). Keep up the good work and eventually you will catch up. If you havent got the Diplomacy ribbon activated, do so, its great help to track whats going on and who you want to be friends/enemies with.
Gilgamesh has a low production start, but he compensates with advancing the techs and civics faster than most, and having one of the most powerful unique units relevant to the era, especially considering they can be built from turn 1, and are very useful tools for taking cities and barb camps.
Take advantage of the small things, however inconsequential they might seem at first. Every bit counts.
Edit;
Key things for Science Victory are Science per turn (duh!

) and production. You want to have at least 1 city. Usually the capital, sometimes the 2nd or 3rd city are better located and have better pottential for production in the way of tiles to place your aqueduct and dam, with the industrial zone in between. Upgrade that industrial zone up to coal power plant, and as soon as you research spaceports, start building it in that city. You can accelerate this process by promoting Moksha or Reyna, and straight up buying the Spaceports with gold or faith, provided your economy allows it. Then start with the projects, and if you can, build/purchase one or 2 spaceports more in similarly developed cities. This will allow you to speed up the last stage when the spaceship is travelling through space, as it is the only one that must be done more than once, and shave several turns too. The government plaza has a building (Royal Society) which allows you to spend builder charges to speed up these projects.
Make sure to place a Spy to defend your industrial zone and spaceport from saboteurs. Best way is to send your first spies to foreign cities to complete easy missions, so they can grab some promotions, then take them back home and place them counterspying on loop. Take into consideration, that a spy is not a guarantee of immunity from sabotage, they can still mess your stuff, but at least you will capture/kill those spies and lower the chances of future sabotage. If you know whose spies they are, send one on offensive missions while working on the spaceport projects, so you can force them to use some for counterspy, and keep that spy (or spies) jumping around random cities delivering some chaos. Extra annoying points for breaching a dam and destroying the districts and improvements of several cities at once
As for districs, a Campus in every city is the "optimal" play, but its way overkill for prince, you can have a very decent science output with just 3 or 4, enough to carry you reliably to the end of the tech tree. You can skip theater squares if you want, enough Ziggurats in your lands give all you need. Commercial hubs and harbors are mostly used for the trade routes, depending on the government you use and the cards you pick, you can get between 5 and 10 production from EACH. Just think how much it can help your main city race through the space projects. You dont need encampments other than for defense, or increasing your strategic resource stock limit, a couple are a good idea, but dont rush them too soon in the game if you dont really need them. You are going to need a couple holy sites too, to build up your faith for the district purchases, also, no need to rush them unless you want a religion real bad.
And wonders... you dont really need them. There are a couple that help, like the Mausoleum at Halicarnassus, which will give your great engineers the ability to use twice their ability (very useful for upgrading your industrial zones and speed up the space projects). The rest are mostly boosts for your science output, and great scientist generation, which campuses already do. So theres that, they help, but they are not necessary.
Just consider that all these optional things do help you win faster, but building them very early in the game will set you back, or expose you to an AI attack because you didnt have time to build a small army. The art is in evaluating when you can do these things succesfuly. The more times you try, the better you will get at it. Myself here, have been a Civ fan since my balls were still bald back in 1992, and have always played prince. Its only now with civ 6 that I have started to try to improve my gameplay. Im doing a game with each civ and watching my victory times come shorter and shorter most of the time, and can finally gather the courage and strength to survive a deity game. Although considering I always play epic speed or marathon speed, Deity turn 15 5x warrior rushes are a hard thing to counter (being very euphemistic here) when all you had time to build was a scout. So I mostly stick to King-Immortal, depending on my familiarity with the civ.
Sorry for the mega wall of text. Got a bit carried away, I hope it helps at least

Ill be happy to clear any doubts, me or any other more experienced people will surely help