I spent a number of years as a plastic fabricator/machinist type, and found that turning acrylic is best done with a large nose radius on the lathe tool, something on the order of .06-.12" radius or thereabout, ground onto a HSS tool, and use the tool then with plenty of relief on the side and front and under the cutting lip, 10-15 degrees or so. With a broad tool nose radius, you can achieve an almost polished finish on a finish pass on the lathe. It's also important to be working with zero rake angle (scraping the material, not digging in to create a chip like metals). This can also apply to twist drills if you have holes to drill.
A strong air blast also is very effective at cooling the cut and getting the chip away. As noted by others, heat is your enemy. I always work dry on machining operations as long as chip control is good, but if you need to do fine finish polishing, wet sanding with 220->400->600 grit SiC wet-dry paper will get you to the point where some light buffing with white compound and a loose cotton buff will be all that's required for cosmetics.
If you need the part to be long-lasting and as durable as it can be, do not flame-polish, but go with sanding and buffing where polishing is necessary. The flame polishing will create a surface layer of highly-stressed material, just waiting for an opportunity to relieve itself by crazing. A wipe with any sort of solvent-based cleaner will often be the trigger for almost instantaneous reaction. Don't ask me how I know. The vapor-polish process is also a possibility, but I have never actually performed it. I have always been curious about the likelihood of crazing with the solvent attack on the surface, but not curious enough to get set up to do it, with all the vapor control in place.
As also noted, if the material is extruded rather than cast, temp problems will be even more apparent. Difficult to get a really good finish on extruded material. Crazing possibilities are multiplied about 5X. A good lubricant is mineral oil if you are working so that you can use it effectively.