Ottomans can stay fairly competitive in tech, even without Westernizing. It doesn't hurt, but it is by no means a requirement. If you are falling far behind it is because of your research strategy or your economy.
Pick the two or three tech groups that are most important to you. Put all your research into one of them until it is finished; that gets you get the benefit for completion as quickly as possible. Then, to avoid the penalty for researching ahead of time, move on to one of the other techs you are focusing on. You'll learn which levels in each tech group are the most important as you play. You should put extra focus on a group when you near one of those important levels.
Lagging more than a few levels behind in Land tech is an easy way to get your armies beat, making it a priority for anyone planning to do much fighting. Naval tech deficits can be overcome by shear numbers (no attrition at sea), making it a common one to skip. Government tech is very important for getting your top ideas and your government of choice; it becomes less important after that. Improving your trade is one of the best ways to speed up your research, but you might have to skip it if you let your reputation get out of hand. The number of groups you cycle through depends on your research rate. Focus on the most important one or two if you are doing very badly, so you can at least keep them up.
Mostly though, you improve your research by improving your economy. Try to take lots of rich provinces, ideally from your culture group. Provinces get a penalty to income if they are a different culture or different religion. Provinces with both penalties will earn you very little. Your research costs are based on the size of your country, so underperforming provinces will drag you down. Convert provinces to your religion right away, and try to limit the number of poor ones you take from outside your culture group. Core provinces give an income bonus though, so always grab them up. Provinces will become core after 50 years, but if they are really crap they might not be worth the wait; prefer taking one really rich province instead of two or three poor ones.
Keep inflation as low as you can. Build workshops as soon as you can, and try to be competitive in the richest centers of trade. Keep your stability at three as often as possible. Pick national ideas that help your trade and taxation. As others have pointed out, the sliders can help both research specifically and your economy in general. Westernization can help as well, though the Ottomans can get by pretty well without it.
Again, it mainly comes down you your economy. If you build a rich nation, you'll keep up in tech. If you don't, you won't.
Of course it takes awhile to actually build that good economy, so don't worry too much about it in the beginning. You are reasonably competitive as long as you are within about three, and you don't have to be competitive in everything. There is nothing stopping the Ottomans from leading the tech race, but you can lag behind a bit and still be successful.
If I missed something please feel free to correct me, if only for my own benefit.
