Can a PP pilot fly large planes?
In the US, yes. There are no restrictions from the FAA on the size of aircraft you fly with a private pilot certificate. If the aircraft requires a type rating, you would have to get that added to your certificate along with the category and class. The insurance carrier covering that aircraft would probably have much more stringent requires over and above the FAA requirements.
John Travolta has/had a Boeing 707 (similar to a B737) that he flies out of his private home/airport.

§61.31 Type rating requirements, additional training, and authorization requirements.
(a) Type ratings required. A person who acts as a pilot in command of any of the following aircraft must hold a type rating for that aircraft:
(1) Large aircraft (except lighter-than-air).
(2) Turbojet-powered airplanes.
(3) Other aircraft specified by the Administrator through aircraft type certificate procedures.
(b) Authorization in lieu of a type rating. A person may be authorized to operate without a type rating for up to 60 days an aircraft requiring a type rating, provided—
(1) The Administrator has authorized the flight or series of flights;
(2) The Administrator has determined that an equivalent level of safety can be achieved through the operating limitations on the authorization;
(3) The person shows that compliance with paragraph (a) of this section is impracticable for the flight or series of flights; and
(4) The flight—
(i) Involves only a ferry flight, training flight, test flight, or practical test for a pilot certificate or rating;
(ii) Is within the United States;
(iii) Does not involve operations for compensation or hire unless the compensation or hire involves payment for the use of the aircraft for training or taking a practical test; and
(iv) Involves only the carriage of flight crewmembers considered essential for the flight.
(5) If the flight or series of flights cannot be accomplished within the time limit of the authorization, the Administrator may authorize an additional period of up to 60 days to accomplish the flight or series of flights.
(c) Aircraft category, class, and type ratings: Limitations on the carriage of persons, or operating for compensation or hire. Unless a person holds a category, class, and type rating (if a class and type rating is required) that applies to the aircraft, that person may not act as pilot in command of an aircraft that is carrying another person, or is operated for compensation or hire. That person also may not act as pilot in command of that aircraft for compensation or hire.
(d) Aircraft category, class, and type ratings: Limitations on operating an aircraft as the pilot in command. To serve as the pilot in command of an aircraft, a person must—
(1) Hold the appropriate category, class, and type rating (if a class or type rating is required) for the aircraft to be flown; or
(2) Have received training required by this part that is appropriate to the pilot certification level, aircraft category, class, and type rating (if a class or type rating is required) for the aircraft to be flown, and have received an endorsement for solo flight in that aircraft from an authorized instructor.