Several possible causes...
All downstream of the toilet: Check your tank vent for a blockage first. A blocked vent pressurizes the tank, which creates backpressure that has to get out somewhere...and the toilet is the least line of resistance. Don't use the toilet again or attempt to pump out the tank till you've either cleared the vent or eliminated it as the cause...'cuz a blocked vent/pressurized tank can cause several disasters a lot worse than just ripe "burps"--even a cracked tank.Most vent blockages occur in the thru-hull and/or in the fitting on the tank...usually due to waste that spills into the vent while heeled and building up. But insects and just dust and pollen can easily clog up a vent thru-hull that has only a tiny slit in it.If anything has gone down the toilet except waste and/or quick-dissolve TP, there could be a partial clog in the head discharge line somewhere...but if you are only getting "burps," and not having any real trouble flushing, I doubt it.Since it's a new boat, I'd also make sure that the vent line or the head discharge hose doesn't have a kink in it. Installers (builders are even more guilty of this than yards) rarely bother to swap out a straight toilet discharge fitting for a 90 or insert an inline fitting to go around a tight bend...they just force the hose. The attached photo illustrates the result.You shouldn't be able to leave an inch or two of water in the bowl, either...marine toilets aren't designed to hold water. The pumping action brings in flush water (or not, if you're in the dry mode) and pumps out the bowl more or less simultaneously. You might want to check out the link below to learn how toilets work, and how to operate and maintain holding tanks and toilets.